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department  of  Ballistics  of  the  U.  S.  Artillery  School. 


EXTERIOR  BALLISTICS 


IN  THE 


PLANE  OF    FIRE 


BY 


JAIVIKS    ISA,    INOAIvIvS, 

Cai'tain  P'ikst  Aktilleky,  U.     S.  Army, 
Instructor, 


NEW    YORK: 
D.    VAN    NOSTRAND,    PUBLISHER, 

23  MURRAY  AND  27  WARREN  STREETS, 
1886, 


■I^*'" 
^V' 


HEADQUARTERS  UNITED   STATES  ARTILLERY  SCHOOL. 

Fort  Monroe,  Va.,  February,  1885. 
Approved  and  Authorized  as  a  Text- Book. 

Pat.  26,  Regulations  U.  S.  Artillery  School,  appioved  1882,  viz.: 

"  To  the  end  that  the  school  shall  keep  pace  with  professional  progress,  it 
is  made  the  duty  of  Instructors  and  Assistant-Instructors  to  prepare  and 
arrange,  in  accordance  with  the  Programme  of  Instruction,  the  subject-matter 
of  the  courses  of  study  committed  to  their  charge  The  same  shall  be  sub- 
mitted to  the  Staff,  and,  after  approval  by  that  body,  the  matter  shall  become 
the  authorized  text-books  of  the  school,  be  printed  at  the  school,  issued,  and 
adhered  to  as  such."  _  -,      ^y 

By  order  of  Lieutenant-Colonel  Tidball. 


Tasker  H.  Bliss, 
First  Lieutenant  ist  Artillery,  Adjutant. 


Copyright,   1886, 
By  D.  van  NOSTRAND. 


PREFACE. 


This  work  is  intended,  primarily,  as  a  text-book  for 
the  use  of  the  officers  under  instruction  at  the  U.  S. 
Artiller}^  School,  and  the  arrangement  of  the  matter  has 
been  made  with  reference  to  the  wants  of  the  class-room. 
The  aim  has  been  to  present  in  one  volume  the  various 
methods  for  calculating  range-tables  and  solving  impor- 
tant problems  relating  to  trajectories,  which  are  in  vogue 
at  the  present  day,  developed  from  the  same  point  of 
view  and  with  a  uniform  notation.  The  convenience  of 
this  is  manifest. 

It  is  hoped,  also,  that  the  practical  artillerist  will  find 
here  all  that  he  may  require  either  for  computing  range- 
tables  for  the  guns  already  in  use,  or  for  determining 
in  advance  the  ballistic  efficiency  of  those  which  may 
be  proposed  in  the  future. 


ERRATA 


Page  54,  line  27  : 

For  -  read  -. 

u  V 

Page  64,  line  4  : 

For  (i)  and  {(f)  read  {i\  and  (^X 

Page  72,  line  18: 

4  i 

For  sec    ^  read  sec  5  f.  / 


Page  73,  line  22  : 


'  -^4-  ^^^^  V* 


Page  93,  line  11  : 

For  g  read  j. 

Page  116,  equation  {78): 

For r—  read 


cos''  (p  2  cos  ip 


CONTKNTS 


INTRODUCTION. 


Object  and  Definitions, 


•AGE 

5 


CHAPTER  I. 

RESISTANCE   OF   THE    AIR. 

Normal  Resistance  to  the  Motion  of  a  Plane, 

Oblique  Motion,         ....... 

Pressure  on  a  Surface  of  Revolution,  .... 

Applications,   ....... 

Resistance  of  the  Air  to  the  Motion  of  Ogival-headed  Projectiles, 

CHAPTER  II. 

EXPERIMENTAL    RESISTANCE. 

Notable  Experiments,  .... 

Methods  of  Determining  Resistances, 

Russian  Experiments  with  Spherical  Projectiles, 

Mayevski's  Deductions  from  the  Krupp  Experiments, 

Ilojel's  Deductions  from  the  Krupp  Experiments, 

Bashforth's  Coefficients, 

Law  of  Resistance  deduced  from  Bashforth's  K,     . 

Comparison  of  Resistances, 

Example,  ..... 


7 

9 

9 

10-13 

13-16 


17 

19 
23 
28 

29 
31 

35 
37 
39 


CHAPTER  III. 

DIFFERENTIAL    EQUATIONS    OF   TRANSLATION — GENERAL    PROPERTIES    OF 
TRAJECTORIES. 

Preliminary  Considerations, 

Notation,         ...... 

DifFerential  Equations  of  Translation, 

Minimum  Velocity,   ..... 

Limiting  Velocity,      ..... 

Limit  of  the  Inclination  in  the  Descending  Branch, 

Asymptote  to  the  Descending  Branch, 

Radius  of  Curvature,  .... 


4.1 
41 
42 
46 

47 

48 

49 
50 


CONTENTS. 


CHAPTER  IV. 

RECTILINEAR    MOTION. 

Relation  between  Time,  Space,  and  Velocity, 
Projectiles  differing  from  the  Standard, 
Formulas  for  Calculating  the  T-  and  ^'-Functions, 
Ballistic  Tables,         .... 
Extended  Ranges,      .... 
Comparison  of  Calculated  with  Observed  Velocities, 


PAGE 

52 
53 
54 
57 
59 
60 


CHAPTER  V. 

RELATION    BETWEEN   VELOCITY    AND    INCLINATION. 

General  Expressions  for  the  Inclination  in  Terms  of  the  Velocity 

Bashforth's  Method,    . 
High-Angle  and  Curved  Fire, 

Siacci's  Method, 

Niven's  Method, 

Modification  of  Niven's  Method, 

CHAPTER  VI. 

HIGH-ANGLE   FIRE. 

Trajectory  in  Vacuo, 
Constant  Resistance, 
Resistance  Proportional  to  the  First 

Euler's  Method, 

Bashforth's  Method,    . 

Modification  of  Bashforth's  Method, 


Power  of  the  Velocity, 


64 

65 
66 
68 
73 
75 


77 
80 
81 
91 
95 
97 


CHAPTER  VII. 

DIRECT    FIRE. 

Niven's  Method, 

. 

102 

Sladen's  Method, 

.          106 

Siacci's  Method, 

.          108 

Practical  Applications, 

.118 

Correction  for  Altitude, 

.          127 

EXTERIOR  BALLISTICS 

IN  THE  PLANE  OF  FIRE. 


INTRODUCTION. 


Definition  and  Object. — Ballistics,  from  the  Greek 
l^aUw,  I  throw,  is,  in  its  most  general  signification,  the 
science  which  treats  of  the  motion  of  heavy  bodies  pro- 
jected into  space  in  any  direction ;  but  its  meaning  is  usu- 
ally restricted  to  the  motion  of  projectiles  of  regular  form 
fired  from  cannon  or  small  arms. 

The  motion  of  a  projectile  may  be  studied  under  three 
different  aspects,  giving  rise  to  as  many  different  branches 
of  the  subject,  called  respectively  Interior  Ballistics,  Ex- 
terior Ballistics,  and  Ballistics  of  Penetration, 

1.  Interior  Ballistics.— Interior  Ballistics  treats  of 
the  motion  of  a  projectile  within  the  bore  of  the  gun  while 
it  is  acted  upon  by  the  highly  elastic  gases  into  which  the 
powder  is  converted  by  combustion.  Its  object  is  to  deter- 
mine by  calculation  the  velocity  of  translation  and  rotation 
which  the  combustion  of  a  given  charge  of  powder  of 
known  constituents  and  quality  is  capable  of  imparting  to 
a  projectile,  and  the  effect  upon  the  gun. 

2.  Exterior  Ballistics. — Exterior  Ballistics  considers 
the  circumstances  of  motion  of  a  projectile  from  the  time 
it  emerges  from  the  gun  until  it  strikes  the  object  aimed 
at.  Its  data  are  the  shape,  caliber,  and  weight  of  the  pro- 
jectile, its  initial  velocity  both  of  translation  and  of  rotation. 


6  EXTERIOR   BALLISTICS. 

the  resistance  it  meets  from  the  air,  and  the  action  of  grav- 
ity. 

3.  Ballistics  of  Penetration. — This  branch  of  the 
subject  has  reference  to  the  effect  of  the  projectile  upon 
an  object;  the  data  being  the  energy  and  incHnation  with 
which  the  projectile  strikes  the  object,  the  nature  of  the  re- 
sistance it  encounters,  etc. 

The  above  is  not  the  order  in  which  the  three  divisions 
of  the  subject  are  usually  presented  to  the  practical  artil- 
lerist, but  the  reverse.  He  desires  to  penetrate  or  destroy 
a  given  object — say  the  side  of  an  armored  ship.  Ballistics 
of  penetration  enables  him  to  determine  the  minimum  en- 
ergy which  his  projectiles  must  have  on  impact,  and  the 
proper  striking  angle,  to  accomplish  the  desired  result. 
Exterior  Ballistics  would  then  carry  the  data  from  the  ob- 
ject to  be  struck  to  the  gun,  and  determine  the  necessary 
initial  velocity  and  angle  of  elevation.  Lastly,  Interior 
Ballistics  would  ascertain  the  proper  charge  and  kind  of 
powder  to  be  used  to  give  the  projectile  the  initial  velocity 
demanded. 

The  following  pages  treat  only  of  Exterior  Ballistics; 
and  this  subject  will  be  limited,  at  present,  to  motion  in  the 
vertical  plane  passing  through  the  axis  of  the  piece. 


CHAPTER   I. 

RESISTANCE   OF   THE  AIR. 

Preliminary  Considerations. — The  molecular  the- 
ory of  gases  is  not  yet  sufficiently  developed  to  be  made 
the  basis  for  calculating  the  resistance  which  a  projectile 
experiences  in  passing  through  the  air.  We  know,  how- 
ever, that  if  a  body  moves  in  a  resisting  medium,  fluid  or 
gaseous,  the  particles  of  the  fluid  must  be  displaced  to  allow 
the  body  to  pass  through  ;  and  hence  momentum  will  be 
communicated  to  them,  which  must  be  abstracted  from  the 
moving  body.  From  the  assumed  equality  of  momenta 
lost  and  gained  Newton  deduced  the  law  of  the  square  of 
the  velocity  to  express  the  resistance  of  the  air  to  the  mo- 
tion of  a  body  moving  in  it. 

The  following,  which  is  the  ordinary  demonstration, 
supposes  the  particles  of  air  against  which  the  body  im- 
pinges to  be  at  rest,  and  takes  no  account  of  the  reaction  of 
the  molecules  upon  each  other,  nor  of  their  friction  against 
the  surface  of  the  body.  The  result  will  therefore  be  but  an 
approximation,  which  must  be  estimated  at  its  true  value  by 
means  of  well-devised  and  accurately-executed  experiments. 

Normal  Resistance  to  the  Motion  of  a  Body 
presenting  a  Plane  Surface  to  the  Medium.— Let 
a  moving  body  present  to  the  particles  of  a  fluid  against 
which  it  impinges,  and  which  are  supposed  to  be  at  rest,  a 
plane  surface  whose  area  is  5,  and  which  is  normal  to  the 
direction  of  motion.  Let  w  be  the  weight  of  the  moving 
body,  V  its  velocity  at  any  time  t,  d  the  weight  of  an  unit- 
volume  of  the  fluid,  and  ^  the  acceleration  of  gravity.  The 
plane  5  will  describe  in  an  element  of  time  dt  a  path  v  d  t, 
and  displace  a  volume  of  fluid  Svdt ;  therefore  the  mass 

of  fluid  put  in  motion  during  the  element  of  time  is-  Svdt. 


8  EXTERIOR   BALLISTICS. 

And  as  this  moves   with  the  velocity  v,  its  momentum   is 

—  Sv'dt;  and  this  has  been  abstracted   from  the  moving 

body,  whose  velocity   has  thereby  been  decreased  by  dv. 
Therefore 

dvzm-Sv'dt 

g  g 

or  IV  dv       b  ^   ^ 

~  -  -y-  =  -  5  z/" 
g  dt      g 

The  first  member  of  this  last  equation  is  the  momentum- 
decrement  of  the  body,  due  to  the  pressure  of  the  fluid 
upon  the  plane  face  5,  and  is  therefore  a  measure  of  this 
pressure.     Calling  this  latter  P,  we  have 


_,  w  dv       <5    -    , 

g  dt        g 


or,  per  unit  of  mass, 

w  dt       w 

As  before  stated,  several  circumstances  have  been  omit- 
ted in  this  investigation  ^vhich,  if  taken  into  account,  would 
probably  increase  the  pressure  somewhat,  at  least  for  high 
velocities.  We  will  therefore  introduce  into  the  second 
member  of  the  above  equation  an  undetermined  multiplier 
k  {k  y  i),  and  we  have 

P  =  k-Sv'  .. 

g  (0 

The  pressure  is,  therefore,  proportional  to  the  area  of 
the  plane  surface,  to  the  density  of  the  medium,  and  to  the 
square  of  the  velocity. 

If  in  equation  (i)  we  make  5=1,  the  second  member 
will  then  express  the  normal  pressure  upon  an  unit-surface 
moving  with  the  velocit}^  v;  calling  this /o,' we  have 

and 


P^AS 


EXTERIOR  BALLISTICS. 


Oblique  Motion. — If  the  surface  5  is  oblique  to  the 
direction  of  motion,  let  f  be  the  angle  which  the  normal  to 
the  plane  makes  with  that  direction  ;  and  resolve  the  velo- 
city 7.'  into  its  components  v  cos  f,  perpendicular,  and  v  sin  f, 
parallel,  to  vS.  This  last,  neglecting  friction,  having  no  re- 
tarding effect,  we  have  for  the  normal  pressure  upon  5  the 
expression 

P=/^-'z^''5cos' 6=/o  5cos'f  / 

Poncelet  {Mecanique  Industrielle,  403)  cites  the  following 
empirical  formula  for  calculating  the  normal  pressure,  viz.  : 

i"fsec'  e  ^  ' 

derived  by  Colonel  Duchemin  from  the  experiments  of 
Vince,  Hutton,  and  Thibault.  As  this  expression  satisfied 
the  whole  series  of  experiments  upon  which  it  was  based 
better  than  any  other  that  was  proposed,  we  will  adopt  it  in 
what  follows. 

Pressvire  on  a  Surface  of 
Revolution. — Let  A  D  B,  Fig. 
I,  be  the  generating  curve  of  a 
surface  of  revolution,  which  we 
will  suppose  moves  in  a  resisting 
medium  in    the    direction   of  its 
axis,^_(9  A.     \{  m  in'  in"  =  <^5  be 
an  element  of  the  surface,  inclined  ^ 
to  the  direction  of  motion  by  the 
angle  Ninv=^e,  it  will  suffer  a 
pressure  in  the  direction  of  the 
normal  N in,  equal,  by  (2),  to 
2p,dS 
I  +  sec'  e 

Resolving  this  pressure  into  two  components. 


2  p^d  S  cos>  8 


2p^d  Ssin  s 


i°+sec'.'  P^""^'^"''  ^"^  i+^c^'  Pe'-Pe"dic"la'-. 


lO  EXTERIOR   BALLISTICS. 

to  OA,  it  is  plain  that  this  last  will  be  destroyed  by  an 
equal  and  contrary  pressure  upon  the  elementary  surface 
n  n'  n"  situated  in  the  same  meridional  section  as  ;;/  in'  m" ,  and 
making  the  same  angle  with  the  direction  of  motion.  It  is 
only  necessary,  therefore,  to  consider  the  first  component, 

2p^d  S  cos  £ 
I  +  sec'  e 

It  is  evident  that  expressions  identical  with  this  last  are 
applicable  to  every  element  of  the  zone  ;//  m'  n  n'  described 
by  the  revolution  of  m  in' ;  and  we  may,  therefore,  extend 
this  so  as  to  include  the  entire  zone  by  substituting  its  area 
for  dS.  If  we  take  O  A  for  the  axis  of  A'',  this  area  will  be 
expressed  by  2  it  yds,  in  which  ds  is  an  element  of  the  gene- 
rating curve ;  therefore,  the  pressure  upon  any  elementary 
zone  will  be 

,  y  ds  cos  f 

dx' 
Substituting  —  dy  for  ds  cos  f ,  and  2  -|-  -r-,  for  i  -\-  sec*  e,  and 

integrating  between  the  limits  4-  =  /,  and  x^o,  we  have 

/'    y<iy 

As.  all  service  projectiles  are  solids  of  revolution,  this 
last  equation  may  be  used  to  calculate  the  relative  pressures 
sustained  by  projectiles  having  differently  shaped  heads,  sup- 
posing their  axes  to  coincide  with  the  direction  of  motion  at 
each  instant.  In  appl3nng  the  formula,  y  will  be  eliminated 
by  means  of  the  equation  of  the  generating  curve.  The 
superior  limit  of  integration  (/)  will  be  the  length  of  the 
head.     R  will  denote  the  radius  of  the  projectile. 

Application  to  Conical  Heads. — Let  /^  i?  be  the  length 
of  the  conical  head,  the  angle  at  the  point  being 


2  tan 


■(0 


EXTERIOR   BALLISTICS. 

The  equation  of  the  generating  line  is 


II 


y=--^R 


whence 


y  dy 


i+i 


and,  therefore, 


d^  n\2-^fe) 

~df 


{n  R  —  x)  dx 


47tp, 


=  7tR'p, 


R 


x)dx 


When  n=:o,  the  head  becomes  flat,  and  the  above  equa- 
tion reduces  to 

P^nR'p, 
as  it  should. 

Application  to  a  Prolate  Hemi-Spheroidal  Head, 
with  Axes  in  the  Ratio  of  one  to  two. — The  equation 
of  the  generating  ellipse  is 

4/  +  ;l;'  =  47?^ 
whence 

y  dy 

x^  dx 


4(8i^^ 


and,  therefore,  since  /  =  2  7?, 

p—  111  I      ^'^-^ 

:=7tR^P,{2\0g2-l) 

Application  to  Ogival  Heads.    J^i 

—Let  A  B  D  (Fig.  2)  be  a  section  of   ^ 
an  ogival  head  made  by  a  plane  pass- 
ing through  the  axis  of  the  projectile. 
Let  A  O  =  Rbe  the  radius  of  the  pro- 
jectile, and  A  £  =  7t  R  be  the  radius 


.\UJ 


12  EXTERIOR   BALLISTICS. 

of  the  generating  circle,  whose  equation  is,  if  we  make  O  the 
origin  and  O  B  the  axis  of  X, 

y^ire  R^-x'Y-in-  \)R 
Making  j^^o,  we  find 

O  B  =  l=:R  V2n  ^i 
Let  the  angle  A  E B=iy  ;  therefore 


V2n—  I 

tan  y  = 

n—  I 

which   serves   to   determine   the   length   of   the  arc  of  the 
ogive,  A  B. 

The  differential  of  the  equation  of  the  generating  circle 
is 

,  X  dx 

^^~~  {it'R'-xY 
whence 

,  J     ,    (n  —  i)  Rx  dx 

and 

,    ,  dx""      n^  R'-X-x' 


dy"  2  x" 

therefore 

^RV^;r:r,  I         2{n—  i)Rx'  2  x'      ) 

^=-2^A/^  I  {n'R'+x'){n'R'-xy  ~  n'R'+x  j 

„,       (       ,    n(n—i)  .       ^  +  V2+  I 
(  1/2  «—  V2+^ 

-«Mog^'+^:-'} 

=  7rR^AF{n),{say)  (3) 

If  «  is  the  angle  at  the  point  of  the  projectile,  the  expres- 
sion for  dj/  gives 


2  n  —  i\ 
n  —  I     J 


a 


EXTERIOR   BALLISTICS. 


13 


When  n=  i,  A  D  B  becomes  a  semi-circle  and  the  head  a 
hemisphere. 

The  following-  table  gives  the  values  oi  F  (n)^  the  lengths 
of  head  in  calibers,  and  the  angles  at  the  point,  for  integral 
values  of  n  from  i  to  6  : 


n 

Fin) 

LENGTH   OF   HEAD 
(0 

ANGLE  AT   POINT 

I 

0.6137 

0.5000 

180°    00'   00'' 

2 

0.4187 

0.8660 

120°   00'   00'' 

3 

0.3176 

I.II8O 

96°    22'   46'' 

4 

0.2560 

1.3229 

82°   49'   09'' 

5 

0.2146 

1.5000 

73°  44'  23'' 

6 

0.1848 

1.6583 

6f    &  ^2" 

Resistance  of  the  Air  to  the  Motion  of  Ogival- 
heacled  Projectiles. — The  expression 

P=7zR'p,F(n) 
which,  by  substituting  for/^  its  value,  becomes 

g 
serves  to  determine  the  pressure,  as  deduced  by  the  above 
theory,  upon  an  ogival  head ;  and  requires  that  this  pressure 
should  be  proportional  to  the  density  of  the  air,  to  the  area 
of  the  cross-section  of  the  body  of  the  projectile,  and  to  the 
square  of  the  velocity.  The  truth  of  the  first  two  of  these 
deductions  may  be  considered  as  fully  established  by  expe- 
riment, and  is  admitted  by  all  investigators.  The  relation 
between  the  front  pressure  and  the  velocity  has  not  been 
satisfactorily  determined  by  experiment,  and  we  are  there- 
fore unable  to  verify  directly  the  law  of  the  square  deduced 
above.  It  seems  probable,  however,  from  experiments  made 
to  determine  the  resistance  of  the  air  to  the  motion  of  pro- 


14  EXTERIOR   BALLISTICS. 

jectiles,  as  well  as  from  theory,  that  this  law  is  approxi- 
mately true  for  all  velocities. 

If  we  represent  the  pressure  of  the  air  upon  the  rear 
part  of  the  projectile  by  P' ,  and  the  resistance  by  />,  we  shall 

evidently  have  ~ 

p^P-P' 

It  is  evident  that  P'  will  be  zero  whenever  the  velocity 
of  the  projectile  is  greater  than  that  of  air  flowing  into  a 
vacuum.  In  this  case,  and  also  when  P'  is  so  small  rela- 
tively  to  P  that  it  may  be  neglected,   we    have   approxi- 

mately 

p^P 

Application  to  Ogival  Heads  struck  witli  Radii 
of  one  and  a  half  Calibers. — Experiments  have  proven 
that  for  practicable  velocities  exceeding  about  1300  f.  s.  the 
resistance  of  the  air  is  sensibly  proportional  to  the  square  of 
the  velocity ;  and  a  discussion  of  the  published  results  of 
Professor  Bashforth's  experiments  has  shown  that,  within 
the  above  limits,  the  resistance  to  elongated  projectiles 
having  ogival  heads  struck  with  radii  of  one  and  a  half  cali- 
bers may  be  approximately  expressed  by  the  equation, 

pz=z-   d'v' 

S 
in  which  d  is  the  diameter  of  the  projectile  in  inches,  ^  the 
acceleration  of  gravity  (32.19  ft.),  and  log  A  =  6,1525284  — 
10.     Whence 

p  —  o.o'44i37^'z;' 

Making  b  =  534.22  grains,  which  is  the  weight  of  a  cubic 
foot  of  air  adopted  by  Professor  Bashforth,  and  F{n)=:F{^) 
=  0.3176,  we  find  for  the  corresponding  expression  for  P 

P  =  o.o%io6g  k  d' v' 

A  comparison  of  the  second  members  of  these  two  equa- 
tions seems  to  warrant  the  conclusion  that  for  velocities 
greater  than  about  1300  f.  s.,  the  rear  pressure  is  either  zero 
or  so  small  relatively  to  the  front  pressure  that  it  may  be 


EXTERIOR   BALLISTICS.  1 5 

neglected  without  sensible  error.     Equating  the  two  mem- 
bers, we  find  for  velocities  greater  than  1300  f.  s. 

k—  1.0747 

In  the  following  table  the  first  and  second  columns  give 
the  velocities  and  corresponding  resistances,  in  pounds,  to 
an  elongated  projectile  one  inch  in  diameter  and  having  an 
ogival  head  of  one  and  a  half  calibers.  They  were  deduced 
from  Bashforth's  experiments  by  Professor  A.  G.  Greenhill, 
and  are  taken  from  his  paper  published  in  the  Proceedings 
of  the  Royal  Artillery  Institution,  No.  2,  Vol.  XIII.  The 
third  column  contains  the  corresponding  pressures  upon  the 
head  of  the  projectile  computed  by  the  formula 

576^ 
in  which  the  constants  have  the  values  already  given.     The 
fourth  and  fifth  columns  are  sufficiently  indicated  by  their 
titles. 

These  results  are  reproduced  graphically  in  Plate  I. 
A  is  the  curve  of  resistance  (^),  drawn  by  taking  the  velo- 
cities for  abscissas  and  the  corresponding  resistances,  in 
pounds,  for  ordinates.  This  curve  is  similar  to  that  given 
by  Professor  Greenhill  in  his  paper  above  cited.  B  is  the 
curve  of  front  pressures  (P),  and  is  a  parabola  whose  equa- 
tion is  given  above.  It  will  be  seen  that  while  the  velocity 
decreases  from  2800  f.  s.  to  1300  f  s.,  the  two  curves  closely 
approximate  to  each  other;  the  differences  (P— />)  for  the 
same  abscissas  being  relatively  small  and  alternately  plus 
and  minus.  As  the  velocity  still  further  decreases,  the  curve 
of  resistance  falls  rapidly  below  the  parabola  B,  showing 
that  the  resistance  now  decreases  in  a  higher  ratio  than  the 
square  of  the  velocity.  This  continues  down  to  about  800 
f.  s.,  when  the  parabolic  form  of  the  curve  is  again  resumed, 
but  still  below  B.  The  differences  P—  p  from  z/=  1300  f.  s. 
to  2/=  100  f  s.  are  shown  graphically  by  the  curve  (7,  which 
may  represent,  approximately,  the  rear  pressures  iox  decreas- 
ing velocities,  and  possibly  account,  in  a  measure,  for  the 


i6 


EXTERIOR  BALLISTICS. 


sudden  diminution  of  resistance  in  the  neighborhood  of  the 
velocity  of  sound. 


V 

p 

P 

P-P 

P-P 

V 

p 

P 

P-P 

P-P 

P 

P 

2800 
2750 

2700 

35.453 
33.586 
31.846 

34.603 
33.378 
32.176 

-0.850 
-0.208 
+  0.330 

1080 
1070 
1060 

3-999 
3.756 

3.478 

5.148 
5.053 
4.959 

+  1.149 
1.297 
1. 481 

0.223 
0.256 
0.298 

2650 
2600 
2550 

30.241 
28.613 
27.243 

30.995 
29.836 
28.700 

+  0.754 
+  1.223 

+  1.457 

1050 
1040 
1030 

3.139 
2.823 
2.604 

4.866 
4.774 
4.684 

1.727 
1. 951 
2.080 

0.355 
0.409 
0.444 

2500 

2450 
2400 

26.406 

25.898 

25.588 

27.585 
26.493 
25.422 

+  1.379 
+-0.595 
-0.166 

1020 

lOIO 

1000 

2.482 
2.404 
2.330 

4.592 
4.502 
4.414 

2. 114 
2.098 
2.084 

0.459 
0.466 
0.472 

2350 

2300 
2250 

25.242 
24.760 

23.566 

24.374 
23.347 
22.344 

-0.868 
-1. 413 
—  1.222 

990 
980 
970 

2.261 

2.193 
2.127 

4.326 
4.239 
4.153 

2.065 
2.046 
2.026 

0.477 
0.483 
0.488 

2200 
2150 
2100 

22.158 
20.811 
19.504 

21.362 
20.402 
19.464 

-0.796 
-0.409 
—0.040 

960 

950 
940 

2.061 
1.998 
1.935 

4.068 
3.983 
3.900 

2.007 

1.985 
1.965 

0.493 
0.498 
0.504 

2050 
2900 
1950 

18.229 
17.096 
16.127 

18.548 
17.654 
16.783 

+  0.319 

+  0.558 
+  0.656 

930 

920 
910 

1.874 
1. 814 
1.756 

3.817 
3.736 
3.655 

1.943 
1.922 
1.899 

0.509 

0.515 
0.520 

1900 
1850 
1800 

15.364 
14.696 

14.002 

15.934 
15.106 
14.300 

+  0.570 
+  0.410 
+0.298 

900 
850 
800 

1.699 

1. 431 
1. 212 

3.575 
3.189 
2.825 

1.876 
1.758 
1. 613 

0.525 
0.551 
0.580 

1750 

1700 
1650 

13.318 

12.666 
12.030 

13.517 
12.766 
12.016 

+  0.199 
+  0.100 
—0.014 

750 
700 
650 

1.043 
0.905 
0.784 

2.483 
2. 163 
1.865 

1.440 
1.258 
1. 081 

0.580 
0.581 
0.580 

1600 
1550 
1500 

II. 416 
10.829 
10.263 

11.298 

10.604 

9.930 

—0.018 
-0.225 
-0.333 

-0.342 
-0.273 
— 0.141 

600 
550 
500 

450 
400 

350 

0.674 
0.572 
0.473 

0.381 
0.294 
0.221 

1.589 

1.335 
1. 103 

0.894 
0.706 
0  541 

0.915 
0.763 
0.630 

0.513 
0.412 
0.320 

0.576 
0.572 
0.571 

0.574 
0.583 
0.592 

1450 
1400 
1350 

9.622 
8.924 
8.185 

9.280 
8.651 
8.044 

1300 
1250 
1200 

7.413 
6.637 

5.884 

7.459 
6.896 

6.356 

+  0.046 

0.259 
0.472 

0.006 
0.038 
0.070 

300 
250 

200 

0.162 
0.112 
0.072 

0.397 
0.276 
0.177 

0.235 
0,164 
0.105 

0.592 
0.595 
0.591 

II50 

IIOO 

1090 

5.179 
4.420 
4.221 

5.837 
5.340 
5.244 

0.658 

0.920 

+  1.023 

0.113 
0.172 
0.195 

150 
100 

0.040 
0.018 

0.099 
0.044 

0.059 
+  0.026 

0.594 
0.591 

CHAPTER   11. 

EXPERIMENTAL  RESISTANCE. 

Notable  Experiments. — Benjamin  Robins  was  the 
first  to  execute  a  systematic  and  intelligent  series  of  experi- 
ments to  determine  the  velocity  of  projectiles  and  the  effect 
of  the  resistance  of  the  air,  not  only  in  retarding  but  in  de- 
flecting them  from  the  plane  of  fire.  He  was  the  inventor 
of  the  ballistic  pendulum,  an  instrument  for  measuring  the 
momenta  of  projectiles  and  thence  their  velocities.  He  also 
invented  the  Whirling  Machine  for  determining  the  resistance 
of  air  to  bodies  of  different  forms  moving  with  low  velo- 
cities. His  *'  New  Principles  of  Gunnery,"  containing  the 
results  of  his  labors,  was  published  in  1742,  and  immediately 
attracted  the  attention  of  the  great  Euler,  who  translated  it 
into  French. 

The  next  series  of  experiments  of  any  value  were  made 
toward  the  close  of  the  last  century  by  Dr.  Hutton,  of  the 
Royal  Military  Academy,  Woolwich.  He  improved  the 
apparatus  invented  by  Robins,  and  used  heavier  projectiles 
with  higher  velocities.  His  experiments  showed  that  the 
resistance  is  approximately  proportional  to  the  square  of 
the  diameter  of  the  projectile,  and  that  it  increases  more 
rapidly  than  the  square  of  the  velocity  up  to  about  1440  f.  s., 
and  nearly  as  the  square  of  the  velocity  from  1440  f.  s.  to 
1968  f.  s. 

In  1839  ^"<^  1840  experiments  were  conducted  at  Metz, 
on  a  hitherto  unprecedented  scale,  by  a  commission  ap- 
pointed by  the  French  Minister  of  War,  consisting  of  MM. 
Piobert,  Morin,  and  Didion.  They  fired  spherical  projec- 
tiles weighing  from  11  to  50  pounds,  with  diameters  varying 
from  4  to  8.7  inches,  into  a  ballistic  pendulum,  at  distances 
of  15,40,65,90,  and   115  metres;  by  this  means  velocities 


I8  EXTERIOR   BALLISTICS. 

were  determined  at  points  25,  50,  75,  and  100  metres  apart, 
the  velocities  varying  from  200  to  600  metres  per  second. 

From  these  experiments  General  Didion  deduced  a  law 
of  resistance  expressed  by  a  binomial,  one  term  of  which  is 
proportional  to  the  square,  and  the  other  to  the  cube,  of  the 
velocity.  This  gave  good  results  for  short  ranges  ;  but  with 
heavy  charges  and  high  angles  of  projection  the  calculated 
ranges  were  much  greater  than  the  observed. 

Another  series  of  experiments  was  made  at  Metz,  in  the 
years  1856,  1857,  and  1858,  by  means  of  the  electro-ballistic 
pendulum  invented  by  Captain  Navez,  of  the  Belgian  Artil- 
lery. This,  unlike  the  ballistic  pendulum,  affords  the  means 
of  measuring  the  velocity  of  the  same  projectile  at  two 
points  of  its  trajectory.  The  results  of  these  elaborate  ex- 
periments may  be  briefly  stated  as  follows:  The  resistance 
for  a  velocity  of  320  m.  s.  does  not  differ  sensibly  from  that 
deduced  from  the  previous  experiments  at  Metz;  but  the 
resistances  decrease  with  the  velocity  below  320  m.  s.,  and 
increase  with  the  velocity  above  320  m.  s.,  more  rapidly  than 
resulted  from  the  former  experiments.  The  commission 
having  charge  of  these  experiments,  whose  president  was 
Colonel  Virlet,  expressed  the  resistance  of  the  air  by  a 
single  term  proportional  to  the  cube  of  the  velocity  for  all 
velocities. 

In  1865  the  Rev.  Francis  Bashforth,  M.A.,  who  had  then 
been  recently  appointed  Professor  of  Applied  Mathematics 
to  the  advanced  class  of  artillery  officers  at  Woolwich, 
began  a  series  of  experiments  for  determining  the  resistance 
of  the  air  to  the  motion  of  both  spherical  and  oblong  projec- 
tiles, which  he  continued  from  time  to  time  until  1880.  As 
the  instruments  then  in  use  for  measuring  velocities  were 
incapable  of  giving  the  times  occupied  by  a  shot  in  passing 
over  a  series  of  successive  equal  spaces,  he  began  his  labors 
by  inventing  and  constructing  a  chronograph  to  accomplish 
this  object,  which  was  tried  late  in  1865  in  Woolwich 
Marshes,  with  ten  screens,  and  with  perfect  success.  It  was 
afterwards  removed   to  Shoeburyness,  where  most  of  his 


EXTERIOR   BALLISTICS.  I9 

subsequent  experiments  were  made.  He  employed  rifled 
guns  of  3,  5,  7,  and  9-inch  calibers,  and  elongated  shot  hav- 
ing ogival  heads  struck  with  radii  of  i^  calibers;  also 
smooth-bore  guns  of  similar  calibers  for  firing  spherical 
shot.  From  the  data  derived  from  these  experiments  he 
constructed  and  published,  from  time  to  time,  extensive 
tables  connecting  space  and  velocity,  and  time  and  velocity, 
which  for  accuracy  and  general  usefulness  have  never  been 
excelled.  The  first  of  these  tables  was  published  in  1870, 
and  his  Final  Report,  containing  coefficients  of  resistance 
for  ogival-headed  shot,  for  velocities  extending  from  2800 
f.  s.  to  JOG  f.  s.,  was  published  in  1880.  These  experiments 
will  be  noticed  more  in  detail  further  on. 

General  Mayevski  conducted  some  experiments  at  St. 
Petersburg,  in  1868,  with  spherical  projectiles,  and  in  the 
following  year  with  ogival-headed  projectiles,  supplement- 
ing these  latter  with  the  experiments  made  by  Bashforth  in 
1867  with  9-inch  shot.  An  account  of  these  experiments, 
with  the  results  deduced  therefrom,  is  given  in  his  "  Traite 
Balistique  Exterieure,"  Paris,  1872. 

General  Mayevski  has  recently  (1882)  published  the  re- 
sults of  a  discussion  of  the  extensive  experiments  made  at 
Meppen  in  1881  with  the  Krupp  guns  and  projectiles. 
These  latter,  though  varying  greatly  in  caliber,  were  all 
sensibl}^  of  the  same  type,  being  mostly  3  calibers  in  length, 
with  an  ogive  of  2  calibers  radius.  General  Mayevski's 
results,  together  with  Colonel  HojeFs  still  more  recent  dis- 
cussion of  the  same  data,  will  be  noticed  again. 
-7?  Methods  of  Determining  Resistances. — If  a  prO'-* 
jectile  be  fired  horizontally,  the  path  described  in  the  first 
one  or  two  tenths  of  a  second  may,  without  sensible  error, 
be  considered  a  horizontal  right  line ;  and,  therefore,  what- 
ever loss  of  velocity  it  may  sustain  in  this  short  time  will  be 
due  to  the  resistance  of  the  air,  since  the  only  other  force 
acting  upon  the  projectile,  gravity^  may  be  disregarded,  as 
it  acts  at  right  angles  to  the  projectile's  motion.  For  ex- 
ample, an  8-inch  oblong  shell,  having  an  initial  velocity  of 


20  EXTERIOR    BALLISTICS. 

1400  f.  s.,  will  describe  a  horizontal  path,  in  the  first  two- 
tenths  of  a  second  after  leaving  the  gun,  of  278  ft.,  while  its 
vertical  descent  due  to  gravity  will  be  less  than  8  inches. 
Moreover,  if  its  velocity  should  be  measured  at  the  distance 
of  278  ft.  from  the  muzzle  of  the  gun,  it  would  be  found  to 
be  but  1380  f.  s.,  showing  a  loss  of  velocity  of  20  f.  s.,  due  to 
the  resistance  of  the  air. 

The  relation  between  the  horizontal  space  passed  over 
by  a  projectile  and  its  loss  of  velocity  may  be  determined 
as  follows  : 

Let  w  be  the  weight  of  the  projectile  in  pounds,  V  and 
V  its  velocities,  respectively,  at  the  distances^  and  a'  from 
the  muzzle  of  the  gun,  in  feet  per  second,  and  g  the  accele- 
ration of  gravity.     The  vis  viva  of  the  projectile  at  the  dis- 

.   wV"  zv  V" 

tance  a  from  the  gun  is ,  and  at  the  distance  a\ : 

^  g 

consequently  the  loss  of  vis  viva  in   describing  the   path 

vu 
a' —a^  is  -( F^—  V  ^) ;  and  this,  by  the  principle  of  vis  viva,  is 

equal  to  twice  the  work  due  to  the  resistance  of  the  air.  If 
the  distance  a'— a  is  not  too  great,  say  from  100  to  300  ft., 
according  to  the  velocity  of  the  projectile,  it  may  be  as- 
sumed that  for  this  distance  the  resistance  will  not  vary 
perceptibly ;  and  if  p  is  the  mean  resistance  for  this  short 
portion  of  the  trajectory,  we  shall  have 

'^{V'-V'^)  =  2{a'-a)p 
whence 

P-     2g{a'-a) 

As,  the  resistance  of  the  air  is  proportional  to  its  density, 
which  is  continually  varying,  it  is  necessary,  in  order  to 
compare  a  series  of  observations  made  at  different  times,  to 
reduce  them  all  to  some  mean  density  taken  as  a  standard. 
If  b  is  the  density  of  the  air  at  the  time  the  observations  are 
made,  and  b^  the  adopted  standard  density  to  which  the  ob- 


/'  = 


EXTERIOR   BALLISTICS.  21 

servations  are  to   be  reduced,  the  second   member  of  the 
preceding  equation  shoidd  be  multipHed  by  ~^   which  gives 

'  2g{a'  —  a)    d 

We  may  take  for  the  value  of  (\  the  weight  of  a  cubic 
foot  of  air  at  a  certain  temperature  and  pressure;  o  will  then 
be  the  weight  of  an  equal  volume  of  air  at  the  time  of  mak- 
ing the  experiments,  as  determined  by  observations  of  the 
thermometer,  barometer,  and  hygrometer. 

As  ft  is  the  mean  resistance  for  the  distance  a^  —  a,  it  may 

VA-V 
be  considered  proportional  to  the  mean  velocity,  v^ — ; 

and  substituting  this  in  the  above  expression,  it  becomes 

wv{V-V')  d,  , 

By  varying  the  charge  so  as  to  obtain  different  values 
for  Fand  V,  the  resistance  corresponding  to  different  ve- 
locities may  be  determined,  and  thence  the  /aw  of  resistance 
deduced. 

In  order  to  compare  the  results  obtained  with  projec- 
tiles of  different  calibers,  the  resistance  per  unit  of  surface 
(square  foot)  is  taken  ;  and,  to  make  the  results  less  sensible 
to  variations  of  velocity,  Didion  proposed  to  divide  the 
values  of  o  by  -J^  and  compare  the  quotients  (p')  instead  of 

/>.     Therefore,  making  ^t  — — ^2-^,  equation  (4)  becomes 

^      gTzF^via'  -a)  3  ^^^ 

It  will  be  observed  that  since  p  is  divided  by  ^'',  the 
values  of  f/  will  be  constant  when  the  resistance  varies  as 
the  square  of  the  velocity  ;  when  this  is  not  the  case  //  will 
evidentl}^  be  a  function  of  the  velocity;  or  f/  =  A' f{v) 
(suppose),  where  the  constant  A',  and  the  form  of  the  fune- 
tion,/(2/),  are  both  to  be  determined. 
3 


/ 


22  EXTERIOR   BALLISTICS. 

Two  assumptions  have  been  made  in  deducing  the  ex- 
pression for  (),  neither  of  which  is  exactly  correct:  ist,  that 
the  resistance  can  be  considered  constant  while  the  pro- 
jectile is  describing  the  short  path  a'  —  a ;  and,  2d,  that  this 
assumed  constant  resistance  is  that  due  to  the  mean  velo- 
city, V.  The  nature  of  the  error  thus  committed  may  be 
exhibited  as  follows: 

The  exact  expression  for  p  is 

w  dv  wv  dv 

^'~~~g~dt~~gds 

Comparing  this  with  (4),  it  will  be  seen  that  we  have  made 

\V-  V  _  _dv 
a'  —  a  ds 

which  is  true  only  when  the  path  described  by  the  projec- 
tile is  infinitesimal. 

To  determine  the  amount  of  error  committed,  we  can  re- 
calculate the  values  of//  by  means  of  the  law  of  resistance 
deduced  from  the  experiments;  and  it  will  be  found  that  in 
the  most  unfavorable  cases  the  two  sets  of  values  of />'  will 
not  differ  from  each  other  by  any  appreciable  amount.  For 
example,  suppose  the  law  of  resistance  deduced  by  this 
method  is  that  of  the  square  of  the  velocity  ;  what  is  the 
exact  expression  for  //  in  terms  of  F~  V  and  a  —  a?  We 
have 

, p      _  w        dv 

^'  ~^:^"J~  '^g^^'  vds 
and  therefore 

,   ,                w     dv 
p'  dsz=L —  — 

whence,  integrating  between  the  limits  Fand  V ,  to  which 
correspond  a  and  a' ,  we  have,  since  p'  is  constant  in  this 
case, 

^'  "^  gTzR^oT^)  ^^^  Y' 
To   test   the   two   expressions   for  //,    take   the   follow 


EXTERIOR   BALLISTICS.  23 

ing  data  from   Bashforth's  ''Final  Report,"  page  19,  round 
486: 

F=2826  f.  s. ;   F'  =  2777  f.  s. ;  7e'  — Solbs. ;  7?  =  4  in.  =  ^ft.; 
F  —  F'  =  49  ;     ^^=  32.191  ;    a'  —  a=^  i$o  ft.,   and  z^  = 

V+  V 


2 


=  2801.5. 


We  find  ■ — ^-i^^-y—, v=:  0.047463;    and    this   is   a   factor  in 

^-rrR'ia  -a) 

both  expressions   for  />'.      Therefore,    by  the  approximate 

method, 

f/  =  0.047463  28^-T  =  0.00083 

and  by  the  exact  method, 

^    1       2826 
f)'  =  0.047463  log =  0.00084. 

For  a  second  example,  suppose  the  law  of  resistance  to 
be  that  of  the  cube  of  the  velocity.  In  this  case  f/  varies  as 
the  first  power  of  the  velocity,  or  f/  =^  A^  v.     Therefore 

A,   1  2v       dv 

^^  TT  R       V 

whence 

II 

^,^      ee.      F^~"F 


gTzK'    a'   -    a 
and 

.'-A'^^-- "^ v{V-V'y 

'     —        "~  gT.k'ia'  -a)         W 

Comparing  this   with  (5),  it  will  be  seen  that  (omitting  the 

factor  ^0   the    two   equations    are    identical,    if   we  assume 

z;^  =  VV  ;  and  this  is  very  nearly  correct  when,  as  in  the 
present  case,  V  —  V  is  very  small  compared  with  either 
For  v. 

As  an  example  of  this  method  of  reducing  observations, 
the  experiments  made  at  St.  Petersburg  in  1868  by  General 


24 


EXTERIOR   BALLISTICS. 


Mayevski,  with  spherical  projectiles,  have  been  selected. 
In  these  experiments  the  velocities  were  determined  by 
two  Boiilenge  chronographs,  and  the  times  measured  were 
in  every  case  within  the  limits  of  o.''io  and  o.''  15. 


X 

* 

< 

•f 

* 

it 

♦ 

\ 

dq\. 

' 

\ 

\> 

§  w 


The  experiments  were  made  with  6  and  24-pdr.  guns 
and  120-pdr.  mortars,  and  the  velocities  ranged  from  745 
f.  s.  to  1729  f.  s.     At  least  eight  shots  were  fired  with   the 


EXTERIOR   BALLISTICS. 


25 


same  charge;  the  value  of//  was  calculated  for  each  shot, 
and  the  mean  of  all  the  values  of />'  so  calculated  was  taken 
as  corresponding  to  the  mean  velocity  of  all  the  shots  fired 
with  the  same  charge.  The  values  o^  a' —  a  varied  from 
164  ft.  to  492  ft.,  the  least  values  being  taken  for  the 
heaviest  charges,  and  the  greatest  values  for  the  smallest 
charges.  The  greatest  loss  of  velocity  {V  —  V)  was  131 
ft.,  and  the  least  33  ft. 

The  values  of  {/  deduced  from  these  experiments  are 
given  in  the  following  table.  For  convenience  English 
units  of  weight  and  length  are  employed ;  that  is,  the 
weights  of  the  projectiles  are  given  in  pounds,  the  veloci- 
ties in  feet  per  second,  and  the  radii  of  the  projectiles  and 
the  values  of  <^'  —  ^  in  feet. 


Values  of  p    for  Si'Herical  Projectiles,  deduced  from  the  Experi- 
ments MADE  AT  St.  Petersburg  in  1868. 


Mean 

Mean 

Kind  of  Gun. 

Velocity 

Values  of 
P' 

Kind  of  Gun. 

Velocity 

Values  of 

6-pdr.  gun 

745  f.  s. 

0.000561 

24-pdr.  gun 

1247  f.  s. 

0.001054 

24-pdr.  gun 

768  " 

508 

0-pdr.  gun 

1260   " 

"45 

120-pdr.  mort. 

860  " 

687 

120-pdr.  mort. 

1339  " 

1117 

6-pdr.  gun 

912  " 

807 

6-pdr.  gun 

1362  " 

1189 

24-pdr.  gun 

942  " 

782 

24-pdr.  gun 

1499  " 

1138 

120-pdr.  mort. 

1083   " 

934 

120-pdr.  mort. 

I5I9  " 

1 163 

24-pdr.  gun 

TII9    " 

987 

6-pdr.  gun 

1558  " 

1189 

6-pdr.  gun 

II22      " 

0.001107 

24-pdr.  gun 

1729  " 

0.001178 

These  results  are  reproduced  graphically  in  Fig.  3,  the 
velocities  being  taken  for  abscissas,  and  the  corresponding 
values  of//  for  ordinates.  It  will  be  seen  that  the  trend  ot 
the  last  seven  points  is  nearly  parallel  to  the  axis  of  ab- 
scissas, and  may,  therefore,  be  represented  approximately 
by  the  right  line  A,  whose  equation  is 

/>'z=  0.00116 

in  which  the  second  member  is  the  arithmetical  mean  of  the 
last  seven  tabulated  values  of  />'. 


26  EXTERIOR   BALLISTICS. 

It   was   found   that  the   remaining  points  could  be  best 

represented    by  a   curve   B,  of  the   second   degree,   of  the 

form  (/  =:  p-\-q  7>^,  containing  two  constants  p  and  q  whose 

values  were   determined   by    the   method   of  least  squares, 

each    tabular   value    of  //  and  the  corresponding    value    of 

V   furnishing  one   "  observation    equation."     it    was    found 

that  the  most  probable  values  of/  and  q  were^/  =  0.012 

and  ^  =  0.00000034686 ;    or,    reducing    to    English    units  of 

k  k 

weight  and  length  by  multiplying  /  by  -  ^,   and  q  by      «, 

where  k  is  the  number  of  pounds  in  one  kilogramme,  and  m 
the  number  of  feet  in  one  metre,  we  have 

^>'  =  0.00022832  -[-0.00000000061309  v" 

or,  in  a  more  convenient  form, 

^/ = 0.00022832  )i+(g-^ J  [ 

To  find  the  point  of  intersection  of  the  right  line  A  with 
the  curve  B,  equate  the  values  of />'  given  by  their  respective 
equations,  and  solve  with  reference  to  v.  It  will  be  found 
that  v^  1233  f.  s.,  at  which  velocity  we  assume  that  the  law 
of  resistance  changes. 

In  strictness  there  is  probably  but  one  laiv  of  resistance^ 
and  this  might  be,  perhaps,  expressed  by  a  very  complicated 
function  of  the  velocity,  having  variable  exponents  and  co- 
efficients, depending,  upon  the  ever- varying  density  of  the 
air,  the  cohesion  of  its  particles,  etc.  ;  but,  however  compli- 
cated it  may  be,  we  can  hardly  conceive  of  its  being  other 
than  a  continuous  function.  But,  owing  to  the  difficulties 
with  which  the  subject  is  surrounded,  both  experimental 
and  analytical,  it  is  usual  to  express  the  resistance  by  in- 
,  tegral  powers  of  the  velocity  and  constant  coefficients,  so 
chosen,  as  in  the  above  example,  as  to  represent  the  mean 
resistance  over  a  certain  range  of  velocity  determined  by 
experiment. 

*  Mayevski,  "  Traite  de  Balistique  Exterieure,"  page  41. 


EXTERIOR   BALLISTICS. 

Expression  for  />. — The  expression  for  />  in  terms  of 

which,  since  [/  is  generally  a  function  of  7',  may  be  written 

The  resistance  per  unit  of  mass,  or  the  retarding  force,  will 
therefore  be 

or,  taking  the  diameter  of  the  projectile  in  inches, 

The  first  member  of  this  equation  expresses  the  retarding 
force  when  the  air  is  at  the  adopted  standard  density  and 
the  projectile  under  consideration  is  similar  in  every  respect 
to  those  used  in  making  the  experiments  which  determined 
//.  To  generalize  the  equation  for  all  densities  of  the  at- 
mosphere we  must  introduce  into  the  second  member  the 

factor  IT  ;  and  we  will  also  assume,  at  present,  that  the  equa- 
tion will  hold  good  for  different  types  of  projectiles  if  d'^  be 
multiplied  by  a  suitable  factor  {c),  depending  upon  the  kind 
of  projectile  used.  For  the  standard  projectile  and  for 
spherical  projectiles,  6=1;  for  one  offering  a  greater  re- 
sistance than  the  standard,  6'>i;  and  if  the'  resistance 
offered  is  less,  r  <  i.     Making,  then, 

576 
and 

^~  d  cd' 
we  have  for  all  kinds  of  projectiles 

p-  dv      A  ^ ,  .  ,^. 

C  is  called  the  ballistic  coefficient,  and  c  the  coefficient  of 
reduction. 


i. 


28  EXTERIOR   BALLISTICS. 

For  the  Russian  experiments  with  spherical  projectiles 
the  standard  density  of  air  to  which  the  experiments  were 
reduced  was  that  of  air  half  saturated  with  vapor,  at  a  tem- 
perature of  1 5°  C,  and  barometer  at  o"'. 75.  In  this  condition 
of  air  the  weight  of  a  cubic  metre  is  1^.206;  and,  therefore, 
the  weight  of  a  cubic  foot  ( =  o)  is  0.075283  lbs.  =  526.98  grs. 
The  value  of  ^  taken  was  9"\8i  =  32.1856  feet.  Applying 
the  proper  numbers,  we  have  the  following  working  expres- 
sions for  the  retarding  force  for  spherical  projectiles. 

Velocities  greater  than  1233  f.  s. : 

^  /?  =  —  7/%-    log  A  =  6.3088473  -  10 
Velocities  less  than  1233  f.  s. : 

f-  r  =  c  '''  V  "^  ?'  /  '  ^^^  ^  ^  5.6029333  -  10 

r  =  612.25  ^^• 

Oblong  Projectiles:  Oeneral  Mayevski's  For- 
mulas.— General  Mayevski,  by  a  method  similar  in  its  gen- 
eral outline  to  that  given  above,  the  details  and  refinements 
of  which  we  omit  for  want  of  space,  has  deduced  the  fol- 
lowing expressions  for  the  resistance  when  the  Krupp  pro- 
jectile is  employed,  viz. :  '^' 

700™  >V>  419™,  />  =  0.0394  TT  R^  -^  v^ 

419'"  >v>  375"\  ^o  =  0.0^94  r  R'  -yv' 


375""  >  -^  >  295"^,  p  —  o.o'67  7:  R"  -^v" 
295^^  >  z;  >  240^ />  =  0.0^583  ;r /?^  y  7^^ 
240™ >  v>  o™,      />  =  0.014 7: T?'^  -^  v" 
Changing  these   expressions  to  the  form  here  adopted 

*  Revue  d^Artilleriey  April,  1883. 


EXTERIOR   BALLISTICS.  29 

[equation  (6)],  and  reducing  to  English  units  of  weight  and 
length,  they  become 

2300  ft.  >  z/>  1370  ft.  : 


ir- 

=  ^T/%-    log  yi  =6.1192437 - 
1370  ft.  >^'>  1230  ft.: 

-  ID 

i'- 

--^  7>\-      log  ^  =  2.9808825  - 

1230  ft.  >  •z/>97o  ft.: 

ID 

ir- 

-  J,  v" ;    log  A  =  6.8018436  - 
970  ft.  >  z/>  790  ft.: 

•20 

i"- 

^     3       , 

790  ft.  >  7'  >  0  ft. : 

•  ID 

i" 

=  ^7^-      log  ^=5.6698755  - 

ID 

Colonel  Hojel's  Deductions  from  the  Krupp  Ex- 
periments.—Colonel  Hojel,  of  the  Dutch  Artillery,  has 
also  made  a  study  of  the  Krupp  experiments  discussed  by 
General  Mayevski :  and,  as  it  is  interesting  and  instructive 
to  compare  the  resistance  formulas  deduced  by  each  of  these 
two  experts,  both  using  the  same  data,  we  give  a  brief  syn- 
opsis of  Colonel  Hojel's  method  and  results. 

He  expresses  the  resistance  by  the  following  formula, 
easily  deduced  from  equation  (6): 

in  which,  from  (4), 

It  is  assumed  that  the  loss  of  velocit}^  V  —  V\  is  some  func- 
tion of  the  mean  velocity  v,  which  can  be  expressed  approx- 
imately, for  a  limited  range  of  velocity,  by  a  monomial  of  the 
form 

4 


30  EXTERIOR   BALLISTICS. 

in  which   A   and   n  are   constants  to  be  determined.     The 
method  of  procedure  is  analogous  to  that  followed  m  deter- 
mining fj',  and  need   not  be  repeated.     Colonel  Hojel  has 
considered   it    necessary    to   employ    fractional   exponents, 
thereby   sacrificing   simplicity   without  apparently  gaining 
in  accuracy.     The  results  he  arrived  at  are  as  follows:  " 
700^  >v>  soo'",    /{v)  =  2A 868  v'-'' 
500™ >  -6^ >  400™,    / (tj)  =  0.29932  z/'" 
400™ >v>  350'",    / (v)  =  o.o'205 524 7/'-'' 
350°^  >  ^  >  300"\    /  (v)  =  o.o'2 1692  V* 
300™  >v>  I40"\    /{z')  =  0.033814  v'-' 
Substituting  these  values  oi /{v)  in  the  equation 

w^         zv    -^      ■'      4w      -^  ^  ^ 
and  reducing  the  results  to  English  units,  that  is,  taking  w 
in  pounds,  v  in  feet,  and  d  in  inches,  we  have  as  the  equiva- 
lents of  Hojel's  expressions,  all  reductions  being  made,  the 

following : 

2300  ft.  >  7^  >  1640  ft. : 

a-         A 
±- p  z=z  ~  v'''\-    log  y4  =6.4211771  —  ID 

1640  ft.  >  7^>  1310  ft. : 

-|.«-^^^-";  iog^  =  5.3923859- 10 

1 3 10  ft.  >  7/ >  II 50  ft. : 

0-         A 

-|-^  =  — 7/^«%-    log ^  =  0.4035263  -  10 

1150  ft.  >7'>98o  ft.: 

a-         A 
^p  =  —v\-      log  ^  =  6.8232495  -  20 

980  ft.  >  7.'  >  460  ft. : 

<r  A 

^  f)=  —  v"-" ;     log  A  —  4.3060287  —  10 

Comparison    of  Resistances  dedviced  from  tlie 
above  Formnlas. — Making  ^=  i  and  f^,  =  o,  in  the  above 

*  Revue  iV Artilleries  June,  1884. 


EXTERIOR   BALLISTICS. 


31 


formulas,  gives  the  resistance  in  pounds  per  circular  inch  at 
the  standard  density  of  the  air.     Calling  this  ^o^,  we  have 

A    ^ 

The  following  table  gives  the  values  of  p^  for  different 
velocities  according  to  Mayevski's  and  Hojel's  formulas  re- 
spectively ;  and  also  the  same  derived  from  "  Table  de 
Krupp,"  Essen,  1881: 


Velocity 
in  feet 
per  sec. 

According 

to 
Mayevski. 

P/ 

According 

to 

Hojel. 

p' 

According 
.  to 
Krupp. 

Velocity 
in  feet 
per  sec. 

P/  . 
According 

to 
Mayevski. 

According 

to 

Hojel. 

p' 

According 

to 

Krupp. 

2300 
2250 
2200 

21.629 
20.699 
19.789 

21.598 
20.710 
19.840 

21.637 
20.643 
19.738 

1250 
1200 
II50 

5.807 

4.899 
3.960 

5.715 
4.888 
4.160 

5-753 
4.904 

3-943 

2150 
2100 
2050 

18,900 
18.031 
17.183 

18.987 
18.153 
17.337 

18.900 
17.962 
17.091 

1 100 
1050 
1000 

3-171 
2-513 
1.969 

3.331 
2.640 
2.068 

3-105 
2.480 
2.044 

2000 
1950 
1900 

16.355 
15.547 
14.760 

16.538 
15-757 
14-995 

16.287 

15-359 
14. 611 

950 
900 
850 

I. 581 
1.344 
1. 132 

1-749 
1.527 
1.324 

1.720 
1.486 
1. 318 

1850 
1800 
1750 

13-993 
13.247 
12.521 

14-250 

13-523 
12.815 

13.929 
I3-I81 
12.500 

800 
750 
700 

0.944 
0.817 
0.712 

1. 138 
0.969 
0.815 

1. 162 
0.983 
0.804 

1700 
1650 
1600 

II. 816 
II. 131 
10.467 

12.125 
11.453 
10.713 

II. 818 
11.059 
10.400 

650 
600 

550 

0.614 
0.523 
0.439 

0.677 
0.554 
0.446 

0.648 

0.514 
0.413 

1550 
1500 
1450 

9.823 
9.199 
8.596 

9.981 
9.277 
8.601 

9-752 
9.126 
8.490 

500 
450 
400 

0.364 
0.294 
0.232 

0.351 
0.270 
0.201 

0.313 

1400 
1350 
1300 

8.014 
7.315 
6.535 

7.954 
7-334 
6.6^1 

7.920 

7.238 
6.445 

i 

Bashforth's  Coefficients. — Professor  Bashforth  adopt- 
ed an  entirely  different  method  from  that  just  developed  to 
determine  the  coefficients  of  resistance,  of  which  we  will 
give  an  outline,  referring  for  further  particulars  to  his 
work,*  which  is  well  known  in  this  country. 

*  "  Motion  of  Projectiles,"  London,  1875  ^n^  1881, 


32  EXTERIOR   BALLISTICS. 

ds 
We  have  v  =:  — ,  whence,  differentiating  and   making  s 

the  equicrescent  variable, 

dv ds  d^i 

'~dt~  'df~ 

dv 
and  this  value  of   -r  substituted  in  (6)  gives 


g_    _  ds  d't  _  /dsV  d'^t  __    ^d't 
w^'~     df     ~~\dt)    ds''~^''  ds' 


From  this  it  follows  that  if  the  resistance  varied  as  the  cube 

of  the  velocity,  —  would  be  constant;  and  we  should  have 
ds"^ 

--,=2^,  (say); 

whence,  integrating  twice, 

t  -=1  bs"  -\-  a  s  -\-  c 

which  is  the  relation  between  the  time  and  space  upon  this 
hypothesis.      When  the  resistance  is  not  proportional  to  the 

cube  of  the  velocity,     -    in  the  equation 

^     ds'  ^ 

—  />  =  — --  ir  —  2b  V 
w  *  ds 

will  be  variable,  and  its  value  must  be  so  determined  by  ex- 
periment as  to  satisfy  this  equation  for  each  value  of  v, 
Bashforth's  method  of  deducing  these  values  is  briefly  as 
follows  : 

Ten  screens  are  placed  at  equal  distances  (150  feet)  apart 
in  the  plane  of  fire,  and  the  exact  time  of  the  passage 
of  a  projectile  through  each  screen  is  measured  by  the 
Bashforth  chronograph.  The  first,  second,  third,  etc.,  dif- 
ferences of  these  observed  times  are  taken,  which  call 
d,,  d,,d,,  etc. 

Let  s  be  the  distance  the  projectile  has  moved  from 
some  assumed  point  to  any  one  of  the  screens,  say  the  first ; 


EXTERIOR   BALLISTICS.  33 

/the  constant  distance  between  the  screens;  and  /,^  /,+/^  /,+2/, 
etc.,  the  observed  times  of  the  projectile's  passing  succes- 
sive screens.  Then  from  a  well-known  equation  of  finite 
differences  we  have 

,    ,    n(n—  1)   .    ,   n(n—  i)in  —  2)   , .  , 

ts.ni=^t,-\- ltd,  +  ^——-^  d, H ^^ ~\\ ^- d,  +  etc. 

1.2  I • -  •  3 

in  which  ;/  is  an  arbitrary  variable.  Arranging  the  second 
member  according  to  the  powers  of//,  we  have 

ts.ni^t,-\-n  \d,  —  \d^-\-\d^  —  -d,\  etc.  ) 
\  2  3  4  / 

-f  etc.,  etc., 

terms  multiplied  by  the  cube  and  higher  powers  of  ^/. 

Since  /  is  a  function  of  s,  we  have  t^—f{s)  and  t,^„i:= 
f{s  +  111).    Expanding  this  last  by  Taylor's  formula,  we  have 

,    dt,   nl  ,   dU,  n'l'  ^ 

whence,  equating  the  coefficients  of  the  first  and  second 
powers  of  71  in  the  two  expansions  of  /^  +  „/,  we  have 

/^^  =  ^-i^,+  i-^3_i^^  +  etc. 
ds  '      2    '    '     3  4 

and 

,„  ^V,         ,         ,    ,    II    -        10    7    1     X 

:r-^^=:^,-^3  +  — <-  — <  +  etC. 

The  first  of  these  equations  gives 
ds  I 


dt,         ^-d,-\d,-^^\d,-\d, 
and  the  second 

7; 

ds-"    ' 

where  i\  is  the  velocity  and  -  />  the  resistance  per  unit  of 

mass  at  the  distance  s  from  the  gun. 


.J,,_-q^^_^^  +  iL^._I|^.  +  etc.) 


34 


EXTERIOR    BALLISTICS. 


As  an  example  take  the  following  experiment  made  with 
a  6.92-inch  spherical  shot,  weighing-  44.094  lbs.,  fired  from  a 
7-inch  gun."^  The  times  of  passing  the  successive  screens 
were  as  follows  : 


Screens. 

Passed  at, 
Seconds. 

d. 

d^ 

^3 

I 

2 . 90068 

8431 

306 

10 

2 

2.98499 

8737 

316 

TO 

•3 

3.07236 

9053 

326 

10 

4 

3. 16289 

9379 

336 

10 

5 

3.25668 

9715 

346 

10 

6 

3.35383 

10061 

356 

I  I 

7 

3-45444 

10417 

367 

II 

8 

3.55861 

10784 

378 

9 

3.66645 

11162 

10 

3.77807 

To  find,  for  example,  the  velocity  at  the  first  screen,  we 
have 


150 


1 1.4  t.  s., 


=  1465.3  f"-  s. 


'      0.08431—^0.00306-1-^0.00010 
and  at  the  seventh  screen 

150 
'      0.10417  —  ^0.003674-^0.00011 

The  retarding  forces  at  the  same  screens  are  as  follows: 

or  V^ 

^  f)^=z  - — ^- (o . 00306  —  o .  oooio)  =  o . ooooooi  3 1 56  Z^,'  =  2<^,  V,^ 

and 

-  Pt=  7 — '-^(0.00367  —  0.00011)  =  0.  ooooooi  5822  z'/  =  2^,  z;/. 

As  these  small   numbers  are  inconvenient  in   practice, 

*  Bashforth,  page  43. 


EXTERIOR   BALLISTICS.  35 

Bashforth  substituted  for  them  a  coefficient  K,  defined  by 
the  equation 

A-=24J(.ooo)'. 

In  the  experiment  selected  above  the  weight  of  a  cubic 
foot  of  air  was  553.9  grains  =  (?,  while  the  standard  weight 
adopted  was  530.6  grains  =  d^.     Therefore  we  have 

(150)  (6.92)  553.9 

and 

j^       0.00356  ^^  ^„ 

A;  = ^\K,—  139.6* 

0.00296     '         ^^ 

That  is  to  say,  when  the  velocity  of  a  spherical  projectile 
is  1811.4  f.  s.,  A"=ii6.i;  and  when  its  velocity  is  1465.3 
f.  s.,  A'=  139.6.  By  interpolation  the  values  of  K,  after 
having  been  determined  for  a  sufficient  number  of  velo- 
cities, are  arranged  in  tabular  form  with  the  velocity  as 
argument. 

Bashforth  determined  the  values  of  K  by  this  original 
and  beautiful  method  for  both  spherical  and  ogival-headed 
projectiles ;  and  for  the  latter  for  velocities  extending  from 
2900  f.  s.  down  to  100  f.  s.  The  experiments  upon  which 
they  were  based  were  made  under  his  own  direction  at 
various  times  between  1865  and  1879,  ^'ith  his  chronograph, 
probably  the  most  complete  and  accurate  instrument  for 
measuring  small  intervals  of  time  yet  invented. 

Law  of  Resistance  deduced  from  Bashforth's 
K. — It  will  be  seen,  by  examining  Bashforth's  table  of  A"  for 
ogival-headed  projectiles,  that  as  the  velocity  decreases 
from  2800  f.  s.  down  to  about  1300  f.  s.,  the  values  of  K 
gradually  increase,  then  become  nearly  constant  down  to 
about  ii3of.  s.,  then  rapidly  decrease  down  to  about  1030 
f.  s.,  become  nearly  constant  again  down  to  about  800  f.  s., 
and  then  gradually  increase  as  the  velocity  decreases,  to  the 

♦  Bashforth's  "  Mathematical  Treatise,"  page  97. 


36  EXTERIOR    BALLISTICS. 

limit  of  the  table.  These  variations  show  that  the  law  of 
resistance  is  not  the  same  for  all  velocities,  but  that  it 
changes  several  times  between  practical  limits.  We  may 
use  Bashforth's  K  for  determining  these  different  laws  of 
resistance  as  follows  : 

We  have  for  the  standard  density  of  the  air, 

^  p  =  2bv'  3=  _  _— —  (7) 

w  ^  zv  (looo) 

and 

from  which  we  get 

,_    S76Kv 
^^       ;r^(ioooy 

The  values  of />'  have  been  computed  by  means  of  this 
formula,  for  ogival-headed  projectiles,  from  if  —  2900  f.  s.  to 
V  =  100  f.  s.,  and  their  discussion  has  yielded  the  following 
results : 

Velocities  greater  than  1330  f.  s. : 
o  A 

^p  —  —,7>\-     log  ^  =:  6.1525284—  10 

1330  f.  s.  > 't^>  II20  f.  s.  : 
^/>  =  -^7^';  log  ^=:  3.0364351  -  10 

1 1 20  f .  s.  >  2/  >  990  f .  s. : 
^/>  =  -^-^"/   log  yi  =  3.8865079 -20 

990  f.  s.  >  7^  >  790  f.  s. : 
fv^^'c'^''    log  ^=2.8754872 -10 

790  f.  s.  >  7/  >  100  f.  s. : 

p-         A 
^pz=.-v\-    log  7^  =  5.7703827- 10 

These  expressions,  derived  as  they  are  from  Bashforth's 


EXTERIOR    15ATXISTICS.  37 

coefficients,  give  substantially  the  same  resistances  for  like 
velocities  as  those  computed  directly  by  means  of  equation 
(7).  The  agreement  between  the  two  for  high  velocities  is 
shown  graphically  by  Plate  I.,  in  which  A  is  Bashforth's 
curve  of  resistance,  while  that  part  of  the  parabola,  B,  com- 
prised between  the  limits  ^^2800  f.  s.  and  7'=  1330  f.  s.,  is 
the  curve  of  resistance  deduced  from  the  first  of  the  above 
expressions.  If,  hovvever,  we  compare  these  expressions 
with  those  deduced  by  Mayevski  or  Hojel  from  the  Krupp 
experiments,  it  will  be  found  that  these  latter  give  a  less 
resistance  than  the  former  for  all  velocities. 

This  is  undoubtedly  due  to  the  superior  centring  of  the 
projectiles  in  the  Krupp  guns  over  the  English,  and  to  the 
different  shapes  of  the  projectiles  used  in  the  two  series  of 
experiments,  particularly  to  the  difference  in  the  shapes  of 
the  heads.  The  English  projectiles,  as  we  have  seen,  had 
ogival  heads  struck  with  radii  of  i|  calibers,  while  those 
fired  at  Meppen  had  similar  heads  of  2  calibres,  and, 
therefore,  suffered  less  resistance  than  the  former  indepen- 
dently of  their  greater  steadiness. 

Comparison  of  Resistances. — Let  f)  and  {>^  be  the  re- 
sistances of  the  air  to  the  motion  of  two  different  projectiles 
of  similar  forms  ;  w  and  zv^  their  weights  ;  5  and  S^  the  areas 
of  their  greatest  transverse  sections;  d  and  d^  their  dia- 
meters ;  and  D  and  D^  their  densities.  Then,  if  we  suppose, 
in  the  case  of  oblong  projectiles,  that  their  axes  coincide 
with  the  direction  of  motion,  we  shall  have  from  (6)  for  the 
same  velocity,  since  5  and  S^  are  proportional  to  the  squares 
of  their  diameters. 


i" 

s 

w 

A       V             ^ 

;     and  ^  =  — 

i''~ 

that  is,  for  the  same  velocity  the  resistances  are  proportional 
to  the  areas  of  the  greatest  transverse  sections,   while  the 
retardations  are  directly  proportional  to  the  areas  and  in- 
5 


38  EXTERIOR    BALLISTICS. 


versely  proportional 
tiles  we  have 

to 

the  ) 

A^eights. 

For 

spher 

ical 

projec- 

.5=i;^^^     S,. 

=  i^ 

^A 

,     7i>- 

=  i;r^^A 

an 

d  w^  = 

:t-;r 

^;^.; 

therefore 

_d^D, 
~  dD 

that  is,  for  spherical  projectiles  the  retardations  are  in- 
versely proportional  to  the  products  of  the  diameters  and 
densities.  This  shows  that  for  equal  velocities  the  loss  of 
velocity  in  a  unit  of  time  will  be  less,  and,  therefore,  the 
range  greater,  cceteris paribus,  the  greater  the  diameter  and 
density  of  the  projectile. 

As  the  weight  of  an  oblong  projectile  is  considerably 
greater  than  that  of  a  spherical  projectile  of  the  same  caliber 
and  material,  it  follows  that  the  retardation  of  the  former 
for  equal  velocities  is  much  less  than  the  latter,  indepen- 
dently of  the  ogival  form  of  the  head  of  an  oblong  projectile 
which  diminishes  the  resistance  still  more.  Indeed,  the  re 
tarding  effect  of  the  air  to  the  motion  of  a  standard  oblong 
projectile,  for  velocities  exceeding  1330  f.  s.,  is  less  than  for  a 
spherical  projectile  of  the  same  diameter  and  weight,  and 
moving  with  the  same  velocity,  in  the  ratio  of  14208  to 
20358.  As  an  example,  if  d  and  w  are  the  diameter  and 
weight  of  a  solid  spherical  cast-iron  shot  which  shall  suffer 
the  same  retardation  as  an  8-inch  oblong  projectile  weighing 
180  lbs.  and  moving  with  the  same  velocity,  we  shall  have, 
since  we  know  that  a  solid  shot  14.87  inches  in  diameter 
weighs  450  lbs., 

,_ (14.87)'  X  180X20358 


and 


^  „      —  29.65  inches 

450X64X  14208  ^    •" 

450  X  (20.65)'  .    ,, 

w—  ^^   .     W,  ^^   —3567  lbs. 
(14.87  '  ^^  ^ 


The  retarding  effect  of  the  air  to  the  motion  of  projectiles 


EXTERIOR   BALLISTICS. 


39 


of  different  calibers  but  having  the  same  initial  velocity  and 
angle  of  projection,  is  shown  graphically  in  Fig. 4,  which  was 
carefully  drawn  to  scale.  A  is  the  curve  which  a  projectile 
would  describe  in  vacuo,  B  that  actually  described  b}^  a 
spherical  projectile  14.87  in  diameter  weighing  450  lbs.,  and 
C  that  described  by  a  spherical  shot  5.9  inches  in  diameter 


weighing  26.92  lbs.  The  initial  velocity  of  each  is  1712.6 
f.  s.,  and  angle  of  projection  30°. 

Example. — Calculate  the  resistance  of  the  air  and  the  re- 
tardation for  a  15-inch  spherical  solid  shot  moving  with  a 
velocity  of  1400  f.  s.  Here  ^=  14.87  in.,  7(^  =  450  lbs.,  and 
A  ~  20358X  iQ-l 

Substituting  these  values  in  equation  (6),  we  have 


and 


dv 
dt 


(14.87)'       20358      , 
^-^-^  X  -^  X  (1400)^ 
32.16  10  ^  -1-     / 

(i4.87r 


450 


X  '-'^^  X  (.400)' 


2743  lbs.. 


196.07  f.  s. ; 


that  is,  at  the  instant  the  projectile  was  moving  with  a 
velocity  of  1400  f.  s.  it  suffered  a  resistance  of  2743  lbs.  ; 
and  if  this  resistance  were  to  remain  constant  for  one  second 
the  velocity  of  the  projectile  would  be  diminished  by  196.07 
ft.  As,  however,  the  resistance  is  not  constant,  but  varies  as 
the  square  of  the  velocity,  it  will  require  an  integration  to 
determine  the  actual  loss  of  velocity  in  one  second. 
We  have  from  (6) 

dt  IV 


40  EXTERIOR   BALLISTICS. 

or 

dv  iV   .     , 

-^,=-       Adt 
whence,  integraling  between  the  limits  F,  7>,  we  have 

Now,  making  V=  1400  and  t=zi,  we  find  v  —  1228  f.  s. ; 
and  the  loss  of  velocit}^  in  one  second  is  1400  —  1228  =  172  ft, 


CHAPTER   III. 

DIFFERENTIAL   EQUATIONS   OF   TRANSLATION — GENERAL 
PROPERTIES    OF   TRAJECTORIES. 

Preliiiiiiiary  Considerations.— A  projectile  fired  from 
a  gun  with  a  certain  initial  velocity  is  acted  upon  during  its 
flight  only  by  gravity  and  the  resistance  of  the  air;  the 
former  in  a  vertical  direction,  and  the  latter  along  the  tan- 
gent to  the  curve  described  by  the  projectile's  centre  of 
gravity.  It  will  be  assumed,  as  a  first  approximation,  that 
the  projectile,  if  spherical,  has  no  motion  of  rotation  ;  and, 
in  the  case  of  oblong  projectiles,  that  the  axis  of  the  pro- 
jectile lies  constantly  in  the  tangent  to  the  trajectory  ;  also 
that  the  air  through  which  it  moves  is  quiescent  and  of  uni- 
form densit}'.  xA.s  none  of  these  conditions  are  ever  fulfilled 
in  practice,  the  equations  deduced  will  only  give  what  may 
be  called  the  normal  trajectory^  or  the  trajectory  in  the  plane 
of  fire,  and  from  which  the  actual  trajectory  will  deviate 
more  or  less  It  is  evident,  however,  that  this  deviation 
from  the  plane  of  fire  is  relatively  small ;  that  is,  small  in 
comparison  with  the  whole  extent  of  the  trajectory,  owing 
to  the  very  great  density  of  the  projectile  as  compared  with 
that  of  the  air. 

Notation.— In  Figure  5,  let  (9,  the  point  of  projection, 
be  taken  for  the  origin  of  rectangular  co-ordinates,  of  which 
let  the  axis  of  X  be  horizontal  and  that  of  F  vertical.  Let 
O  A  be  the  line  of  projection,  and  O  B  E  the  trajectory  de- 
scribed.    The  following  notation  will  be  adopted: 

o  denotes  the  acceleration  of  gravity,  which  will  be  taken 
at  32.16  f.  s. ; 

IV  the  weight  of  the  projectile  in  pounds; 

d  its  diameter  in  inches; 

(p  the  angle  of  projection,  A  O  E ; 


42  EXTERIOR   BALLISTICS. 

V  the  velocity  of  projection,  or  muzzle  velocity  ; 
[/  the  horizontal  velocity  of  projection  =  Fcos  (f ; 

V  the  velocity   of  the  projectile  at  any  point  M  of  the 
trajectory  ; 

3  the  angle  included   between  the  tangent  to  the  curve 
at  any  point  Jf  and  the  axis  of  X,  =  T  M  H  ; 
CO  the  angle  of  fall,  CEO; 
Y 


0  D 

u  the  horizontal  velocity  =  ^'  cos  ^; 

/  the  time  of  describing  any  portion  of  the  trajectory 
from  the  origin  ; 

s  the  length  of  any  portion  of  the  arc,  as  O  in  ; 

X  the  horizontal  range,  O  E ; 

T  the  time  of  flight ; 

ft  the  resistance  of  the  air,  or  the  resistance  a  projectile 
encounters  in  the  direction  of  its  motion,  in  pounds. 

Dift'erential  Equations  of  Translation. — The  ac- 
celeration'^" in  the  direction  of  motion  due  to  the  resistance 

of  the  air  is  — //;  and  the  correspondins^  acceleration  due  to 

gravity   is  ^  sin  /> ;    therefore  the  /<?/<:?/ acceleration  in  the 
direction  of  motion  is  expressed  by  the  equation, 

f  =-^-,-,-sin,>  (8) 

The   velocities   parallel  to  X  and    Y  are,   respectively, 

*  The  term  "acceleration"  is  here  used  for  retardation.      To  avoid  multiplying  terms  re- 
tardation will  be  regarded  as  negative  acceleration. 


EXTERIOR   BALLISTICS.  43 

V  COS  <>and  v  sin  />  ;  and  the  accelerations  parallel  to  the  same 

o  g 

axes  are  --  />  cos  />  and  j^  +  --  p  sin  />. 
Therefore 

^  (^    cos    '>)  ^  g  ,     , 

^     ,, =  -  —  /'  COS  /!/  (9) 

at  zu  '  ^^^ 

and 

d  {v  sin  /5^)  ^. 


dt  "-       -  ' 


-^ ^  f>  sin  ^y 

7£/ 


Performing-  the  differentiations  indicated  in  the  above 
equations,  multiplying  the  first  by  sin  &  and  the  second  by 
cos  />,  and  taking  their  difference,  gives 

— ^  =  -^^cos/V  (10) 

Introducing  the  horizontal  velocity  u  =  7'  cos  />  in  (9)  and 
(10),  and  substituting  for  •_'^  ft  its  value  from  (6),  they  become, 

making/ (7/)  =  7^", 

du A  u""  •  , 

rf7  ""  "~  6'^  cos""-^^^  ^^^^ 

and 

-^^  =  —  a- cos' fi  (12) 


whence,  eliminating  <3^/, 

<^  />  g  C  du 


(13) 


cos"^'<>  ^    ?/**^' 

Symbolizing  the  integral  of  the  first  member  of  (13)  by 
(/>)„,  that  is,  making 

^^^"     J   "cos^'^"^ 

71  k^  C 

and  writing  for  the  sake  of  symmetry,  for  -^,  we  shall 

have 

rdu  k^ 

id-\  =  n  k'  I = h  C 


44 


EXTERIOR   BALLISTICS. 


If  (z)  is   the   value   of  (f"^)   when  //   is   infinite,   we  have 
C=(0;  and  therefore 


whence 


and 


={t\-{»). 


k 


k  sec  (5* 


From  (ii)  we  have 


C       ^  ,  Kidn 

(it  ■-— cos"-'  f>—- 

A  ?/" 

and  this  substituted  in  the  equations 

c/x  =  ?/  d/,      dy  =  //  tan  ^  dt,     ds  =  //  sec  li  dt, 

gives 

^,=  __^cos«-,>_ 

./j/=  --^sln/^cos'-^/^f-, 


^^  = 


From  (12)  w^e  have 


C 


cos" 


,>i^ 


dt=  - 


It      d  /> 


^^    cos''  ^> 


d  tan  ^y 


whence,  as  before, 


dx=^ d  tan  & 

g 


dy 
ds 


tan  b-  d  tan  /V 


—  sec  ^  ^  tan  ^^ 


(14) 
(•5) 


(>-) 


(18) 

(19) 
(20) 

(21) 

(22) 
(23) 
(24) 


EXTERIOR   BALLISTICS. 


45 


Eliminating-  u  from  these  last  four  equations  by  means 
of  (15),  they  take  the  following  eleg-ant  forms : 


it=  ~  ~ 


k_  d  tan  d^  (25) 


, ^  ^tan  d^  (26) 

,  k-     tan  ^y^/ tan /V  (27) 

_  _  /P     sec  d  d  tan  />  (28) 


7?^' w<7r/'j-.— Subject  to  the  conditions  specified  in  the  pre- 
liminary considerations,  equations  (16)  to  (20)  or  (25)  to  (28) 
contain  the  whole  theory  of  the  motion  of  translation  of  a 
projectile  in  a  medium  whose  resistance  can  be  expressed  by 
an  integral  power  of  the  velocity.  Equation  (16)  gives  the 
velocity  in  terms  of  the  inclination;  (18)  and  (19)  or  (26)  and 
(27),  could  they  be  integrated  generally,  would  give  the  co- 
ordinates of  any  point  of  the  trajectory,  while  the  time  would 
depend  upon  the  integration  of  (17)  or  (25).  But,  unfortu- 
nately, the  ''laws  of  resistance"  which  obtain  in  our  atmo- 
sphere do  not  admit  of  the  integration  of  these  equations  ; 
we  are,  therefore,  obliged  to  resort  to  indirect  solutions 
giving  approximations  more  or  less  exact.  Of  these  many 
have  been  proposed  by  different  investigators;  but,  with 
few  exceptions,  they  are  either  too  operose  for  practical  use 
or  not  sufficiently  approximate. 

General  Didion,  in  the  fifth  section  of  his  '*  Traite  de 
Balistique,"  gives  a  full  and  interesting  ri^sumi^  o{  \\\q  labors 
of  mathematicians  upon  this  difficult  problem  up  to  his 
time  (1847),  ^"<^  ^"  the  same  work  gives  an  original  solution 
of  his  own  of  great  value.  Within  the  last  quarter  of  a  cen- 
tury much  has  been  accomplished  to  improve  and  simplify 
6 


46  EXTERIOR   BALLISTICS. 

the  methods  for  calculating  tables  of  fire  and  for  the  solution 
of  the  various  problems  relating  to  trajectories  ;  and  we  will 
endeavor  in  the  following  pages  to  present  such  of  these 
methods  as  are  of  recognized  value,  developed  after  a  uni- 
form plan  and  based  upon  the  preceding  differential  equa- 
tions. 

Oeiieral  Properties  of  Trajectories. — Though  it 
is  impossible  with  our  present  knowledge  to  deduce  the 
equation  of  the  trajectory  described  by  a  projectile,  there 
are  certain  general  properties  of  such  trajectories  which 
may  be  determined  without  knowing  the  law  of  resistance, 
if  we  admit  that  the  resistance  increases  as  some  power  of 
the  velocity  greater  than  the  first,  from  zero  to  infinity; 

whence,  making  —  =  /(^)>  we  shall    have  f  {v)  >  o,   and 

/(x)=  X. 

Variation  of  tlie  Velocity — Miniiniim  Velocity. 

— The  acceleration  in  the  dircctio7i  of  motion  is  [equation  (8)] 

f  =-."T/(^')  +  sin*] 

in  which  — ^sin  d  is  the  component  of  gravity  in  the  direc- 
tion of  motion;  and,  therefore,  whether  the  velocity  is  in- 
creasing or  decreasing  with  the  time  at  any  point  of  the  tra- 
jectory, depends  upon  the  algebraic  sign  of  the  second  mem- 
ber; and  this,  since  f  {v)  \      =  — )  is  considered  positive, 

depends  upon  the  sign  of  sin  d^.  In  the  ascending  branch 
sin  &  is  positive,  and,  therefore,  from  the  point  of  projection 
to  the  summit  the  velocity  is  decreasing.  At  the  summit 
sin  ?^  =  o,  and  at  this  point  gravity,  which  has  hitherto  con- 
spired with  the  resistance  to  diminish  the  velocity,  ceases 
to  act  for  an  instant  in  the  direction  of  motion,  and  then,  as 
sin  d-  changes  sign  in  the  descending  branch,  begins  to  act 
in  opposition  to  the  resistance  ;  that  is,  its  action  tends  to 
increase  the  velocity.  The  component  of  gravity  acting 
perpendicular  to  the  projectile's  motion  (^  cos  d),  and  which 


EXTERIOR   BALLISTICS.  47 

is  a  maximum  at  the  summit,  tends  to  increase  the  in- 
clination in  the  descending  branch,  and  thus  to  increase 
(numerically)  —  sin  &,  until  at  a  certain  point  of  the  de- 
scending branch  where  the  inclination  is  (say)  —  &'  the 
acceleration  of  gravity  in  the  direction  of  motion  has  in- 
creased until  it  just  equals  the  retardation  due  to  the  re- 
sistance of  the  air,  which  latter  has  continually  decreased 
with  the  velocity.  Beyond  this  point,  as  the  component  of 
gravity  in  the  direction  of  motion  still  increases  with  the 
inclination  while  the  resistance  remains  constant  for  an  in- 
stant, the  velocity  also  increases;  and,  therefore,  at  the 
point  where 

w 
the  velocity  is  a  minimum,  and  —  =  o. 

Passing  the  point  of  minimum  velocity,  the  acceleration 
of  gravity  and  the  retardation  due  to  the  resistance  of  the 
air  both  increase;  but  that  there  is  no  maximum  velocity, 
properly  speaking,  may  be  shown  as  follows : 

Differentiating  the  above  expression  for  the  acceleration, 
we  have 

d'^v  '       dv  ^  d& 

and  putting  in  place  of  ^  its  value  from  (10),  we  shall  have 

d'v  ^,.  ^dv   ,     .^-^cos^/> 


df  '*''     '  V/    '  V 

dv 
and  this  is  necessarily  positive  whenever —  ==  o.     The  velo- 
city, therefore,  can  only  be  a  minimum  ;  but  it  tends  towards 

1-        •     •  1  •  ^  P  10  '^ 

a  hmitmp:  value,  viz.,  when  — -  =:  i,  and  //=^ . 

Liinitiiig"  Velocity. — As  the  limiting  velocities  of  all 
service  spherical  projectiles  are  less  than  1233  f.  s.,  we  can 


48 


EXTERIOR   liALLlSTlCS. 


determine  these   velocities  by  means  of  the  expression  for 
the  resistance  given  in  Chapter  II.,  from  which  we  get 


A  (P 

o    zv 


(.+o>) 


:=   I 


where  ^4  =  0.000040048  and  r  =  610.25.      Solving  with   re- 
ference to  7'   we  ore t 


Z'+^^f--' 


which  gives  tlie  limiting  velocity. 

The  following  table  contains  the  limiting  velocities  of 
spherical  projectiles  in  our  service  calculated  by  the  above 
formula  : 


Solid  Shot. 

Inches. 

Lbs. 

Final 
Velocity.  ' 
Feet.      i 

Shells 
Unfilled. 

Inches. 

7(' 

Lbs. 

Final 

Velocity. 

iFeet. 

20-inch 

19.87 

1080 

859  : 

15-inch 

14.87 

330 

1 
726      1 

15-inch 

14.87 

450 

783 

13-inch 

12,87 

216 

682 

13-inch 

12.87 

283 

743 

lo-inch 

9.87 

101.75 

635 

lo-inch 

9.87 

128 

684 

8-inch 

7.88 

45 

561 

i2-pdr. 

4.52 

12.3 

526 

i2-pdr. 

4.52 

8.34 

458 

Limit  of  the  Iiieliiiatioii  of  the  Trajectory  in  the 
Desceiidinj>-  Braiicli.  — We  have  assumed  above  that  tlie 
descending  branch  of  tlie  trajectory  ultimately  becomes 
vertical.     To  prove  this,  take  equation  (10),  viz. : 

and  integrating  from  a  point   of  the  trajectory  where  ^y  =  if 
and  ^  =  o,  we  have 

As  the  velocity  v,  between  the  limits  /  =  o  and  /  =  x  ,  is 


XY  ) 


EXTERIOR   BALLISTICS.  49 

finite  and  continuous,  and  cannot  become  zero,   we   have, 
since  v  is  a  function  of  />, 


where  A^  is  some  value  of  v  greater  than  its  least,  and  less 
than  its  greatest  value  between  the  limits  of  integration. 
As  (^  is  negative  in  the  descending  branch,  the  above 

equation  shows  that,  when  /  is  infinite,  /V  is  equal  to  —  '-. 

2 

From  (24)  we  have 


tan(''  +  ^' 

z^^/i  \4      2 

A   log 


ot/s 

=  —v\ . 

COS  /> 

and, 

therefore 

I,  when 

t  is 

infinite. 

0  c  — 

2 

C0 

cos/y- 

:  K' 

<b-^  — 

tan 

\4     4. 

tan  o 


where   K'  is   some  value   of  -ir  greater  than  its  least,  and 
less    than    its    "freatest    value    between    the    limits    of  inte- 


?5 


»(i+-9.,. 


tan 

gration  ;  and,  as  log ^ —    ^  ^  is  infinite,  so  is  the  arc 

tan  o 

which  conesponds  to  /  =  x  . 

Asymptote  to   the   Descending-  Branch.— As  the 

tangent  to  the  descending  branch  at  infinity  is  vertical,  if  it 
can  be  shown  that  it  cuts  the  axis  of  X  at  a  finite  distance, 
it  is  an  asymptote.  To  determine  this,  take  equation  (22) 
which  ofives 


.'•-=/_'. -'^''  =  ^"'('f 


where  K"  is  a  finite  quantity,  since  v^  is  finite  between  the 
limits  of  integration.     Therefore  the  descending  branch  has 


a  vertical  asymptote. 


50  EXTERIOR   BALLISTICS. 

Radius  of  Curvature. — Designate  the  radius  of  curva- 

ture  by  y.     We  have  by  the  differential  calculus  v  = -^ 

(since  the  trajectory  is  concave  toward  the  axis  ofX);  we 
also  have  ds^=zvdt ;  consequently  y  z=z  —-  — —,  and  therefore 
from  (12) 

y  zzz  —  sec  (J  \ 

g 

The  radius  of  curvature  is  therefore  independent  of  the 
resistance  of  the  air,  and  at  any  point  of  the  trajectory  de- 
pends only  upon  the  velocity  and  the  inclination,  and,  there- 
fore, has  the  same  value  for  the  corresponding  points  of  a 
parabola  described  by  a  projectile  in  vacuo.  The  above  ex- 
pression shows  that  the  radius  of  curvature  decreases  from 
the  point  of  projection  to  the  summit  of  the  trajectory, 
since  v  and  sec  d-  both  decrease  between  those  limits.  Be- 
yond the  summit  v  still  decreases,  but  as  sec  d-  increases  we 
cannot  determine  by  simple  inspection  where  y  ceases  to 
decrease  and  becomes  a  minimum.  Differentiatinof  the  ex- 
pression for  y,  we  have 

dy        2v  sec  d^   dv    ,    7/  ,,  ., 

-TTT  = — — 777  +  —  tan  tJ  sec  o- 

d(>  g        dty       g 

From  (13)  and  (6)  we  have 

d{v  cos  d)        ft 


■fe 


V 


dd-  w 

whence,  differentiating  and  reducing, 

+  sin  & 


dv  ''      f   (^ 

d&        cos  d-  \  zv 


Substituting  this  in  the  expression  for   -^  gives 


dy        zr 
—^  =  —  sec 
^'^        g 


^.>g  +  3sin,y) 


This  equation  shows  that  beyond  the  summit  --~r  is  posi- 


EXTERIOR   BALLISTICS.  5  I 

tive  up   to  the    point  where    —    +  3   sin   d  =r  o,  and    then 

changes  its  sign.      At  this  point,    therefore,  the  radius  of 

curvature  becomes  a  minimum  and  afterwards  increases  to 

infinit}^ 

At  the  point  of  maximum  curvature  we   have,  in  conse- 

20 
quence  of  the  condition  ^^  +  3  sin  &  zzz  o, 

-777  = V  tan  If 

dt^  2 

and  therefore,  since  0-  is  negative  in  the  descending  branch, 

-777-  is  positive  at  that  point,  and  v  is  decreasing  with  d-] 
au 

in  other  words,  the  velocity  has  not  yet  become  a  mini- 
mum. Therefore  the  point  of  maximum  curvature  is  near- 
er the  summit  of  the  trajectory  than  the  point  of  minimum 
velocity. 


CHAPTER  IV. 

RECTILINEAR    MOTION. 

Relation  between  Time,  Space,  and  Velocity. — 

For  many  practical  purposes,  and  especially  with  the  heavy, 
elongated  projectiles  fired  from  modern  guns,  useful  results 
may  be  obtained  by  considering  the  path  of  the  projectile 
a  horizontal  right  line,  and  therefore  unaffected  by  gravity. 
Upon  this  supposition  f'i-  becomes  zero,  and  equations  (17), 
(18),  and  (20)  become 

C  dv 

and 

n  dx  ^=-  (is  ^=^ — — — 

A    v^-^ 

whence  integrating,  and  making  /  and  ^  zero  when  7^=  F, 
we  have 

t-c\ ^ L__l 

((;/-  \)Av^-'       {n-  \)A  F^'M 
and 

,^r  S  I L_ ^ 

\{n  —  2)Av^-^       {71— 2)  A  F«-^f 

Writing,  for  convenience, 

T  {%>)  for  ; r   ^    ^  -,  and  5  {v)  for  ; ,   .    „_„ 

these  equations  become 

t^C\T{v)-T{.^\       7\  (29) 

and  rji  '  ' 

s=C\S{v)-S{zt)\     '  (30) 

When  n  =  2,  the  above  expression  for  s  becomes  inde- 
terminate.    In  this  case  we  have 


EXTERIOR   BALLISTICS.  53 

,  C   dv 

whence 

s  =  —\\og  V-\ogv  I 
and  therefore,  when  n  =  2, 

Equations  (29)  and  (30)  (or  their  equivalents)  were  first 
given  by  Bashforth  in  his  *'  Mathematical  Treatise,"  Lon- 
don, 1 873.  He  also  gave  in  the  same  work  tables  of  5  (v)  and 
T{7>)  for  both  spherical  and  elongated  shot;  the  former  ex- 
tending from  V  =:  1900  f.  s.  to  v  =  500  f.  s.,  and  the  latter 
from  V  =  1700  f.  s.  to  v  =:  540  f.  s.  In  a  "  Supplement "  to  his 
work  above  cited,  published  in  1881,  he  extended  the  tables 
for  elongated  projectiles  to  include  velocities  from  2900  f.  s. 
to  100  f.  s. 

Projectiles  differing  from  tlie  Standard.  — It  will 
be  seen  that  the  value  of  the  functions  T{v)a.nd  S  (v)  depend 
upon  those  of  v  and  A,  the  former  of  which  is  independent 
of  the  nature  of  the  shot,  while  the  latter  depends  partly 
upon  the  form  of  the  standard  projectile,  which  in  this 
country  and  England  has  an  ogival  head  struck  with  a 
radius  of  ij  calibers,  and  a  body  2^  calibers  long.  The  fac- 
tor 6^  ( or  -^  ~~)  depends  upon  the  weight  and  diameter  of 
\       o    ca  /  • 

the  projectile,  the  density  of  the  air,  and  the  coefficient  c ; 
which  latter  varies  with  the  type  of  projectile  used.  The 
factor^  varies,  therefore,  with  c ;  but  by  the  manner  in  which 
A  and  c  enter  the  expressions  for  /  and  s,  it  will  be  seen 
that  the  results  will  be  the  same  if  we  make  A  constant, 
and  give  to  ^  a  suitable  value  determined  by  experiment  for 
each  kind  of  projectile.  By  this  means  the  tables  of  the 
functions  T{v)  and  5(z/),  computed  upon  the  supposition 
that  ^  =  I,  can  be  used  for  all  types*  of  projectiles.  We 
will  now  show  how  these  tables  may  be  computed  for  ob- 
long projectiles,  making  use  of  the  expressions  for  the  re- 
7 


54  EXTERIOR   BALLISTICS. 

sistance  derived  from  Bashforth's  experiments  given  in 
Chapter  I. 

Oblong  Projectiles,  Velocities  greater  than  1330 

f.  s. — For  velocities  greater  than  1330  f.  s.  we  have  ^^  =  2 
and  log  ^  =6.1525284  —  10;  therefore 

r(.)  =  -iandr(F)=-i, 

or,  since  the  value  of  t  depends  upon  the  difference  of  T{y) 
and  T{V),  we  may,  if  convenient,  introduce  an  arbitrary 
constant  into  the  expression  for  T{v).  Therefore  we  may 
take 

and,  similarly, 

•S  (^)  =  ^  (-  log  r  +  log  0',)  =  ^  log  ^ 

To  avoid  large  numbers  and  to  give  uniformity  to  the 
tables  we  will  determine  the  constants  Q,  and  Q\  so  that 
the  functions  shall  both  reduce  to  zero  for  the  same  value 
of  v;  and  it  will  be  convenient  to  begin  the  table  with  the 
highest  value  of  v  likely  to  occur  in  practice,  which  we  will 
assume  (following  Bashforth)  to  be  2800  f.  s. 

We  therefore  have 


A   \2800  '  ---V 


2800 


I  1...  Q\ 


loR  -S^  =0   G'l  =  2800 
^  2800       ' 

Substituting  the  above  values  of  A,  Q^,  and  Q\  in  the 
expressions  for  T{v)  and  S  (v),  and  reducing,  we  have  for 
velocities  between  2800  f.  s.  and  1330  f.  s. 

T{v)  =  [3.8474716]  -^-  -  2.5137 

and 

S{v)  =155866.12  —  [4.2096873]  log  V. 

The    numbers    in    brackets  are  the   logarithms   of  the  nu- 
merical  coefficients  of    the  quantities    to    which    they^^are 


EXTERIOR   BALLISTICS.  55 

prefixed  ;  and  the  factor  lo^  v  is  the  common  logarithm  of 
z',  the  modulus  being  included  in  the  coefficient. 

Velocities  between  1330  f.  s.  and  1130  f.  s.— For 

velocities  between  1330  f.  s.  and   11 20  f.  s.  we  have  n  —  i 
and  log  A  =  3.0364351  —  10;  therefore,  as  before, 


Arbitrary  Constants. — To  deduce  suitable  values  for 
the  arbitrary  constants  Q^  and  ^'2,  we  must  recollect  that 
the  function  representing  the  resistance  of  the  air  changes 
its  form  abruptly  when  the  velocity  is  1330  f.  s. ;  and  to 
prevent  a  correspondingly  abrupt  change  in  our  table  at 
the  same  point — that  is,  to  make  the  numbers  in  the  table  a 
continuous  series — we  must  give  to  g^  and  Q^  such  values 
as  shall  make  the  second  set  of  functions  equal  in  value  to 
the  first  when  z;=i330.  They  will,  therefore,  be  deter- 
mined by  the  following  relations: 

_L/     L_  -  ^)  =  lf_J L_^ 

2 A  V(i33o)'  ^  ^7       A  V1330      2800/ 
and 

I  /    I       I    ^/  \        I    1       2800 

in  which  the  A  in  the  first  member  must  not  be  confounded 
with  that  in  the  second.  Making  the  necessary  reductions, 
we  have 


and 


Tizi)  =  [6.6625349]  A-+  0.1791 


V 


5(z/)=  [6.9635649]-^  -  1674.: 


Velocities  between  1130  f.  s.  and  990  f.  s.— For 

velocities  between   1120  f.   s.   and  990  f.   s.  we  have  ;/ =  6 
and  log  A  —  3.8865079  —  20  ;  therefore 


^(-^  =  5i-(^+^") 


56  EXTERIOR    BALLISTICS. 

and  .  V 

The  constants  must  be  determined  as  before,  by  equating 
the  above  expressions  to  the  corresponding  ones  in  the  case 
immediately  preceding,  making  ^=  1120.  The  results  are, 
all  reductions  being  made, 

r(T^)  =  [15.4145221]  ■;:^  + 2.3705 

and 

5(2;)=[i5.5ii432i]  I-  +  4472.7 

Velocities  between  990  f.  s.   and   790  f.  s.— For 

velocities  between  990  f.  s.  and  790  f.  s.  we  have  //  =  3  and 
log  A  =  2.8754872  —  10;   whence 


and 


Proceeding  as  before,  we  have 

r(2;)  =  [6.8234828]  i,- 1.6937 


and 


*^(^)  =  [7-i245i28]  i-  5602.3 


Velocities  less  than  790  f.  s. — For  velocities  less 
than  790  f.  s.  we  have  « 1=  2  and  log  ^4  =  5.7703827  —  10 ; 
therefore 


^(-)=z(^  +  a) 


and 

whence,  as  before, 

T{v)  =  [4.2296173]  -^  -  12.4999 

S{v)  —  124466.4—  [4-59' 8330]  log  2/. 


EXTERIOR   BALLISTICS.  57 

Ballistic  Tables. — Table  I.  gives  the  values  of  the  time 
and  space  functions  for  oblong  projectiles,  computed  by  the 
above  formulas,  and  extends  from  v  zizz  2800  f.  s.  to  v  =  400 
f  s.  The  first  differences  are  given  in  adjacent  columns; 
and  as  the  second  differences  rarely  exceed  eight  units  of 
the  last  order,  it  will  hardh^  ever  be  necessary  to  consider 
them  in  using  this  table. 

Table  II.  gives  the  values  of  these  functions  for  spherical 
projectiles,  and  is  based  upon  the  Russian  experiments  dis- 
cussed in  Chapter  II. 

EXAMPLES  OF  THE  USE  OF  TABLES  I.  AJ^D  II. 

Example  i. — The  velocity  of  an  8-inch  service  projectile 
weighing  180  lbs.  was  found  by  the  Boulenge  chronograph 
to  be  1398  f.  s.  at  300  ft.  from  the' gun.  What  was  the 
muzzle  velocity  ? 

Here   C  ==  ^,  v  =  1398,  and  s  =  300,  to  find  V.      From 
64 

(30)  we  have 

and  from  Table  1.  yXW"* 

5(1398)  =4903.8  -  ^^^^^'^=4888.7 

also 

s  64  ^ 

C  =  300  X-^=  106.7 

whence 

5(F)  =  4782.0 

.  •.  F=  141 5  +  ^1^  =  14^9-4  f-  s. 

Example  2. — Determine  the  remaining  velocity  and  the 
time  of  flight  of  the  12-inch  service  projectile,  weighing  800 
lbs.,  at  1000  yds.  from  the  gun,  the  muzzle  velocity  being 
1886  f.  s. 


58  EXTERIOR    BALLISTICS. 

1.  Fand  s  are  given,  to  find  v;  where  </=  12,  2£/ =  800 

800 

F=  1886,  J  =:  3000,  and  C ^^ 

144 

We  have 

5(.)  =  5(,886)  +  32^^4 

From  Table  I., 

5  (1886)  =  2803.7  -  0.6  X  37.4  =  2781.3 
3000-X  144^ 
800  ^^ 

,    10  X  27.0  ^     ^ 

.  •.  z/=  1740  H —  =  1746.7  f.  s. 

^  40.3 

2.  Fand  z/  are  given,  to  find  //  from  Table  I., 

T{v)=  1.5 16 
r(F)=i.2i7 

r(z;)-  r(F)  =  0.299 

.-./  =  0.299  X  ^=1^66 
144 

"     Example  3. — Suppose  we  wish  to  determine  the  value 

of  the  coefficient  of  reduction,  c,  for  a  particular  projectile 

whose  form  differs  from  the  standard  projectile.     From  (30) 

we  have 

^ w   s 

u  Td}''  S(v)-S(V) 

whence  ^  ^  ^    ^ 

jv  S{v)-S{V) 
d-"  s 

It  is,  therefore,  only  necessary  to  measure  the  velocity 
of  the  projectile  at  two  points  of  its  trajectory  as  nearly  in 
the  same  horizontal  line  as  practicable,  and  at  a  known  dis- 
tance apart,  and  substitute  the  values  thus  obtained  in  the 
above  formula.  For  example,  the  40-centimetre  (71 -ton) 
Krupp  gun  fires  a  projectile  weighing  17 15  lbs.  with  a 
muzzle  velocity  of  1703  f.  s.  By  experiment  it  is  found 
that  the  velocity  at  1800  ft.  from  the  gun  is  1646  f  s.  What 
is  the  value  of  c  for  this  projectile  ? 


EXTERIOR  BALLISTICS. 


59 


Here 

F=  1703,  V  ==  I 

646,  ^  ==  1 80 

15.748. 

From  Table  I., 

^(^)  =  3742.2 

5(F)  3=  3499-7 

log  242.5 

=  2.3846580 

-    w 

=  0.8397959 

clogs 

==  6.7447275 

2t/=  1715,  and  d 


log  ^  =  9.9691814     ^  =  0.9315 
.-.  log  (7=  0.87061451 

Extended  Ranges. — For  the  heaviest  elongated  pro- 
jectiles, fired  with  high  initial  velocities,  the  remaining 
velocities  and  times  of  flight  may  be  determined  by  this 
method  with  sufficient  accuracy  for  quite  extended  ranges; 
that  is  to  say,  for  ranges  due  to  an  angle  of  projection  of 
10°  or  12°,  or,  in  other  words,  when  the  least  value  of  cos  ^ 
for  the  entire  trajectory  does  not  depart  very  much  from 
unity,  its  assumed  value. 

Example  4. — Compute  the  remaining  velocities,  with  the 
data  of  the  last  example,  at  1800  ft.,  3600  ft.,  5400  ft,  ...  up 
to  18000  ft.  from  the  gun. 

The  work  may  be  arranged  as  follows: 

S{v)  =  3499-7»     log  6^=  0.8706145. 


J 

c 

Siv) 

V 

Computed  by 
Krupp's  Formula. 

1800  ft. 
3600  - 
5400 '' 

7200  " 

242.47 
484.9 
727-4 
969-9 

3742.2 
3984.6 
4227.1 
4469.6 

1645  f.  s. 
1589  '' 
1536  - 
1484  " 

1646  f. 
1590   ' 
1536   * 
1484   ' 

S. 

9000  " 
10800  '' 
12600  '' 

I2I2.3 

1454.8 
1697.3 

4712.0 

4954-5 
5197.0 

1434  " 
1385  '' 
1338  " 

1434   ' 
1385  ' 
1338  ' 

14400  '' 
16200  " 

1939.8 
2182.2 

5439-5 
5681.9 

1293  " 
1250  " 

1293   * 
I25I   ' 

18000  " 

2424.7 

5924.4 

1211  *' 

I2II  " 

60  EXTERIOR   BALLISTICS. 

The  numbers  in  the  second  column  are  simple  multiples 
of  the  first  number  in  that  column;  those  in  the  third  column 
are  found  by  adding  S  {V)  =  3499.7  to  the  numbers  on  the 
same  lines  in  the  second  column,  and  the  velocities  in  the 
fourth  column  are  taken  from  Table  I.  with  the  argument 
S{v). 

The  velocities  in  the  last  column  were  computed  by 
Krupp's  formula.  They  are  copied,  as  also  the  data  of  the 
problem,  from  "  Professional  Papers  No.  25,  Corps  of  En- 
gineers, U.  S.  A.,"  page  41. 

In  this  example  the  angles  of  projection  and  fall  for  a 
range  of  18000  feet  are,  respectively,  7°  18'  and  9°  20';  while 
an  8-inch  shell  weighing  180  lbs.  would  require  for  the  same 
range,  with  the  same  initial  velocity,  an  angle  of  projection 
of  11°  5^  and  the  angle  of  fall  would  be  19°  40'. 

In  this  latter  case  the  velocity  computed  by  the  above 
method  would  not  be  a  very  close  approximation. 

Comparison  of  Calculated  with  Observed  Velo- 
cities,— The  following  table,  taken,  with  the  exception  of 
the  last  two  columns,  from  "Annexe  a  la  Table  de  Krupp," 
etc.,  Essen,  1881,  shows  the  agreement  between  the  observed 
and  calculated  velocities  for  projectiles  having  ogives  of  2 
calibers.  The  sixth  column  gives  the  distances,  in  metres, 
between  the  points  at  which  the  velocities  were  measured 
{X^  and  X^).  The  seventh  and  eighth  columns  give  the 
observed  velocities  at  the  distances  from  the  gun  X,  and  X^ 
respectively.  The  ninth  column  gives  the  velocities  at  the 
distances  X^  from  the  gun  computed  by  Krupp's  table  and 
formula.  The  tenth  column  gives  the  velocities  at  the  dis- 
tances X^  computed  by  equation  (30),  using  Table  I.  of  this 
work.  The  coefficient  of  reduction  (c)  was  taken  at  0.907, 
which  is  its  mean  value  for  velocities  between  2300  f.  s.  and 
1200  f.  s.,  as  determined  by  a  comparison  of  Bashforth's  and 
Krupp's  tables  of  resistances  given  in  Chapters  I.  and  II. 
The  only  discrepancies  of  any  account  between  the  calcu- 
lated velocities  in  this  column  and  the  observed  velocities 
occur  when  the  curvature  of  the  trajectory  is  considerable, 


EXTERIOR   BALLISTICS. 


6i 


JU 

-H.S 

c« 

rt 

. 

i 

0 

•§^ 

tn 

>, 

>, 

& 

V 

u 
>»  / 

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C 

s 

6 

s 
s 

B 

1 

0 

^  0  c< 

53 

Is 

r 

■0  c 

2 

} 

1  '^ 

a 
0 

S 
0 

I 

240 

2.8 

125 

1-245 

1450 

467 

380 

379.9 

380.7 

380.6 

2 

240 

2.8 

161 

1.245  1450   1 

454.5 

390 

388.3 

•387.7 

387.5 

3 

172.6 

2.8 

61.5 

1.226 

1389 

477 

388 

388.7 

389.3 

388.7 

4 

172.6 

2  8 

61.5 

1.226 

1429 

514.7 

416.6 

417.^ 

417.6 

415.7 

5 

149. 1 

2.8 

39-3 

1.260  1429   1 

518 

401.6 

402.1 

403.0 

401.2 

6 

149. 1 

2.5 

33.5 

1.240 

1429 

507.7 

380 

380.7 

379.9 

379.1 

7 

149. 1 

2.8 

31-3 

1.265 

924 

475.8 

387.8 

388.2 

387.7 

387.3 

8 

355 

2.8 

525 

1.200 

1884 

495-9 

432.7 

433-1 

433.8 

432.6 

9 

355 

2.8 

525 

1.200 

2384 

490 

415 

411. 8 

414.4 

412.3 

lO 

355 

2.8 

525 

1.200 

2389 

488.5 

409.6 

410.4 

412.3 

410.9 

II 

149. 1 

2.8 

31.3 

1.265 

1950 

609- 

394 

393-9 

395.4 

392.7 

12 

149. 1 

4 

51 

1.206 

1929 

505-2 

394.6 

393.3 

393.4 

392.3 

13 

152.4 

4 

51.5 

1.205 

1450 

472.4 

391.3 

389-3 

389.1 

388.6 

14 

152.4 

2.8 

32.5 

1.205 

1450 

577 

422 

422.0 

424.2 

421.5 

15 

149-1 

2.8 

31.3 

1.230 

1450 

632.4 

460.9 

460.3 

462.8 

459.8 

i6 

240 

3.8 

215 

1.208 

1904 

480.4 

412.8 

412.0 

412.4 

411. 1 

17 

400 

2.8 

777 

1. 180 

2384 

499.4 

433.7 

432.1 

433 -o 

431.7 

i8 

400 

2.8 

643 

1. 190 

2384 

533.4 

443-8 

447.0 

448.2 

446.6 

19 

400 

2.8 

643 

I.I90 

2384 

531.5 

444-5 

445-4 

446.6 

445.0 

20 

84 

2.8 

6.55 

1. 197 

2447 

446.9 

266 

267.2 

259-7 

267.4 

21 

120 

2.8 

16.4 

1. 211 

2447 

463.3 

284.1 

289.2 

281.6 

289.3 

22 

149. 1 

2.8 

31-3 

1.285 

3448 

536.6 

294.8 

290.6 

283.7 

290.5 

23 

105 

3.5 

16 

1.300 

3436 

481.5 

282 

278-4 

271.2 

279.6 

24 

96 

3.5 

12 

1.340 

3439 

425.8 

256.2 

.250.5 

244.1 

254.4 

25 

107 

2.7 

12.5 

I. 218 

777.5 

205.1 

188.2 

189.8 

187.7 

189.8 

20 

152.4 

2.8 

31.5 

1.206 

966.5 

203 

188 

187.4 

185.9 

188.0 

27 

105 

3.5 

16 

1.222 

950 

514.2 

426.9 

421. 1 

422.2 

420.4 

28 

149. 1 

2.8 

39 

I. 218 

1429 

470 

369-5 

370.4 

369.1 

369.3 

29 

283 

2.5 

234-7 

1.206 

4450 

464.7 

321.2 

31S.9 

311-3 

317.6 

30 

283 

2.5 

234-7 

1.205 

|i879 

465-3 

403.9 

403.3 

404.6 

403.7 

31 

283 

2.5 

234.7 

1.200 

11919 

465.9 

385.4 

384.7 

384-0 

383.8 

32 

283 

2.5 

234.7 

1.200 

12425.5 

466.5 

370.6 

368.0 

366.6 

367.0 

33 

283 

2.5 

234-7 

1.220 

2921.5 

464.8 

347-8 

350.9 

347-7 

349.7 

34 

283 

2.5 

234-7 

1.227 

3426.0 

463-7 

336.0 

337.6 

331-4 

336.6 

35 

283 

2.5 

234.7 

1.220 

I4446.5 

460.0 

316.6 

316.6 

308.6 

315.0 

36 

283 

2.5 

234.7 

1. 192 

,5945.0 

1 

455.8 

295.0 

293.9 

285.6 

293-0 

37 

283 

2.5 

234.7 

1.206 

5945.0 

453.1 

294.7 

291.5 

283.2 

291.4 

62  EXTERIOR    BALLISTICS. 

as  in'the  last  four  rounds,  and  one  or  two  others.  Equation 
(30)  is  based  upon  the  supposition  that  the  path  of  the  pro- 
jectile is  a  horizontal  right  line,  and,  of  course,  gives  only 
approximate  results  when  this  path  has  any  appreciable 
curvature.  It  will  be  shown  subsequently  that,  to  obtain 
the  real  velocity,  the  "  v  "  computed  by  (30)  should  be  mul- 
tiplied by  the  ratio  of  the  cosines  of  the  angles  of  projec- 
tion and  fall.  In  No.  37,  for  example,  it  will  be  found  that 
to  attain  a  range  of  5945  metres  (3!  miles)  the  angle  of  pro- 
jection would  have  to  be  12°  37',  and  the  angle  of  fall  would 
be  17°  40'.  Making  the  necessary  correction,  we  should 
find  the  velocity  to  be  290.7  m. 

The  last  column  gives  the  remaining  velocities  computed 
by  Mayevski's  formulas.  They  follow  very  closely  those 
computed  by  Krupp. 

In  the  absence  of  tables  we   ma}^   determine  remaining 

velocities    which    exceed    1300   f.   s.  as  follows:    We  have 

found,  when  n  =  2, 

C  ,        V 
•^  =  -X  log  — 


V        ^  ,    As    ^    UAs^    ,     ^ 

V 


.    As     .    i/AsV   . 


As 
As  -yr  is  usually  a  small  quantity,  all  its  powers  higher 

than  the  first  may  be  neglected,  and  we  may  put 


V  6 


V  As 

V 

1+4 


For  oblong  projectiles  having  ogival  heads  of  i  J-  calibers 
A  ^0.000142.  If  the  ogive  is  of  2  calibers,  A  =0.0001316. 
This  method  gives  correct  results  for  distances  of  a  mile,  or 
even  more,  especially  for  the  heavy  projectiles  used  with 
modern  seacoast  guns.  If  the  data  are  given  in  French  units 
— that  is  zv,  d,  and  d^  in  kilogrammes,  din  centimetres,  and  s 
and  V  in  metres — the  value  of  A  will  be  0.000030357. 


EXTERIOR   BALLISTICS.  63 

Example.  Let  dz=.io.^  cm.,  2e/  =  455  kg.,  <5  =  1.274  kg., 
<5^  =  1.206  kg.,  F=  520.8  m.,  and  .^=1900  m.  [Krupp's 
Bulletin,  No.  31.] 

We  have 

^         455  X  1.206  , 

C  =  ,       \^ =■  0.46301 

(30.5r  X  1.274 

and 

520.8  520.8  . 

0.000030357  X  1900       1. 12457 
^  ~^  0.46301 

The  measured  velocity  in  this  example  was  465.5  m., 
while  the  velocity  computed  by  Krupp  was  460.1  m. 


CHAPTER  V. 

RELATION   BETWEEN   VELOCITY   AND    INCLINATION. 

Expressions  for  the  Velocity. — Equation  (15),  which, 


since  {i)  =  --—  -|-  (^),  may  be  written 


(f)»-(#)»  =  ^-{jr--^4  (31) 

gives  the  relation  between  the  horizontal  velocities  ^and  11 
and  the  corresponding  inclinations  ^  and  d^\  and  of  these 
four  quantities  any  three  being  given,  the  fourth  can  be  ac- 
curately computed,  provided,  of  course,  that  the  value  of  k 
has  been  accurately  determined  by  experiment.  The  func- 
tions (^)„  and  (?^)„  are  the  integrals  of ;^^-^-,  and  the  fol- 

COS         t/" 

lowing  are  the  forms  they  take  for  the  values  of  n  here 
adopted : 

('>),  =  i  {  tan  »  sec  »  +  log  tan  (^  +  y)  } 

{»),  =  tan  »  +  i  tan'  » 


+  A|og.„g+|) 


It  is  evident  that  all  these  expressions  become  o  when 
/>  —  o,    negative    when    f'^   is    negative,    and     infinite    when 

?^  =: '  ;  or,  in  symbols,  (o)  =  o,(—  ^)  =  —  (/5^),  and  r' j  =  x 

If  there  were  buto^ie  "  law  of  resistance" — in  other  words, 
\^  n  had  but  one  value  for  all  velocities — it  would  be  easy  to 
calculate  the  velocity  for  any  given  value  of  />  by  means  of 


EXTERIOR   BALLISTICS. 


65 


(31).  It  would  only  be  necessary  to  tabulate  the  values  of  (/>)„ 
for  all  practical  values  of  d-  as  the  argument,  and  to  pro- 
vide a  similar  table  of  (-j   with  ?/ as  the  argument.     But,  as 

we  have  seen,  ;i  may  change  its  value  two  or  three  times  in 
the  same  trajectory  ;  and  though  it  would  be  possible  to 
ascertain  by  trial  the  exact  point  of  the  trajectory  where 
this  change  occurred,  yet  the  labor  involved  would  be  very 
great. 

Basliforth's  Method. — Professor  Bash  forth  overcomes 
this  difficulty  by  giving  to  71  the  constant  value  3,  and 
making  /r'  to  vary  in  such  a  manner  as  to  satisfy  (31)  for  all 
velocities.  His  method  of  procedure  is  as  follows:  making 
«  =  3  and  />  =  o,  (31)  becomes 


C/' 


i^tan  ^  +  i  tan>^ 


in  which  6^  and  (f  are  the  horizontal  velocity  and  inclination, 
respectively,  at  the  beginning  of  an}^  arc  of  the  trajectory 
we  may  be  considering;  and  v^  the  velocity  at  the  summit. 
In  Bash  forth 's  notation 


3^        ^' 


^(loooy  w' 


substituting  this  in  the  above  equation  and  multiplying  by 
(1000)^  to  avoid  the  inconvenience  of  very  small  numbers, 
we  have 

/iooo\^       /iooo\'       K  d'  i     ^  ,   .     3     ) 

by  means  of  which  either  z^^,  [/,  or  (p  can  be  determined 
when  the  other  two  are  known.  When  the  resistance  can 
be  taken  proportional  to  the  cube  of  the  velocity,  K  is  con- 
stant; but  for  all  other  velocities  it  is  a  variable,  and  we 
must  take  a  certain  mean  of  its  values  for  the  arc  under  con- 
sideration. Prof.  Bashforth  takes  the  arithmetical  mean, 
which  will  generally  give  very  accurate  results  for  arcs  of 


66  EXTERIOR    BALLISTICS. 

lo  or  15  degrees  in  extent.     In  his  work  he  gives  the  ne- 

cessar}^  tables  for  suitably  determining  —  for  all   velocities 

from  100  f.  s.  to  2900  f.  s.,  and  also  tables  giving  values  of 
3  tan  ip  -\-  tan^  ip  for  all  practical  values  of  ^ . 

Other  approximate  methods  involving  less  labor  will  be 
given  further  on. 

High  Aiig:le  and  Curved  Fire. — When  the  initial 
velocity  does  not  exceed  800  f  s.,  which  includes  nearly  all 
mortar  and  howitzer  practice,  the  law  of  resistance  for 
oblong  projectiles  is  that  of  the  square  of  the  velocity; 
whence,  making  n  ^  2,  and  dropping  the  subscript,  (31)  be- 
comes 

or,  writing  /  (u)  for 


(^)-(^)=^{/W-/(C/))  (32) 

The  value  of /(?/)  for  any  given  value  of  ti  can  be  taken 
directly  from  Tables  T.  and  II.,  the  method  of  construction 
of  which  will  be  given  further  on.  Table  III.  gives  (^)  and 
extends  from  ^  =  o  to  (^  =  60°. 

To  use  (32)  for  computing  low  velocities  (and  also  for 
high  velocities,  exceeding  1330  f.  s.),  we  have 

/«=f  I  (?)-('?)}  + /(f^)  (33) 

2 
in  which  u  and  B  are  the  only  variables;  -^,  (^),  and  I{U)j 

having  been  determined,  do  not  change  their  values  for  the 
same  trajectory. 

To  illustrate  the  ease  with  which  velocities  may  be  cal- 
culated by  (33),  take  the  following  data  from  Bashforth's 
"Treatise,"  page  115: 


EXTERIOR    BALLISTICS. 


67 


V^  751  f.  s. ;    ^  ==  30°;   w  =  70  lbs.,  and  (i'  =  6.27  inches. 
Here  C/==  751  cos  30°  ^  650.385  f.  s. ;    and  from  Table 

I., /(r7)  =  a93354;  -g.  = -^  =  1. 12323. 
We  will,  following  Bashforth,  compute  the  velocities  for 


^  =  28°,    24°,  20^ 


40°.       The  work   may  be  conve- 


niently arranged  as  follows: 

{f)  z=  0.60799        I{U)  =  0.93354. 


e 

(») 

(<<,)  -  (0) 

~({6)  -  (0)) 

/(«) 

(Table  I.) 
u 

«  sec  6  =  V 

Bash- 
forth's 

Differ- 
ence. 

30° 

0.60799 

0 . 00000 

0.00000 

0.93354 

650.38 

751-0 

75I.O 

0.0 

28" 

.55580 

.05219 

.05862 

0.99216 

636.09 

720.4 

720.4 

0.0 

24° 

.45953 

.14846 

.16675 

I . 10029 

612.03 

669.5 

670.2 

-  -7 

20° 

.37185 

.23614 

.26524 

I. 19878 

592.33 

630.3 

630.5 

.2 

16° 

.29063 

.31736 

.35647 

I . 29001 

575.69 

598.9 

598.9 

0.0 

12° 

.21415 

.39384 

•44237 

1.37591 

561.23 

573.8 

573.5 

+  .3 

8° 

. 14100 

.46699 

.52454 

1.45808 

548.38 

553.8 

553.1 

•7 

4° 

. 06998 

.53801 

.60431 

i^537S5 

536.71 

538.0 

537.0 

i.o 

o"^ 

.00000 

.60799 

.68291 

I. 61645 

525-91 

525.9 

524.6 

1.3 

-4" 

—  .06998 

.67797 

.76151 

1.69505 

515.74 

517.0 

515.5 

1.5 

8° 

.14100 

•  74899 

.84129 

1.77483 

505.99 

511.0 

509.3 

1.7 

12° 

.21415 

.82214 

•92345 

1.85699 

496.52 

507.6 

505.7 

1.9 

16° 

. 29063 

.89862 

1.00935 

1.94289 

487.15 

506.8 

504.7 

2.1 

20° 

.37185 

.97984 

.  I. 10056 

2.03410 

477.77 

508.4 

506.2 

2.2 

24" 

•45953 

1.06752 

I . 19906 

2. 13260 

468.22 

*5i2.5 

510.2 

2.3 

28° 

•55580 

I. 16379 

1.30720 

2 . 24074 

458.38 

5i9^i 

516.8 

2.3 

32° 

.66343 

I. 27142 

1.42809 

2.36163 

448.06 

528.3 

525.9 

2.4 

36° 

.78617 

I. 39416 

1.56596 

2.49950 

437.11 

540.3 

537.9 

2.4 

40° 

.92914 

1.53713 

1.72654 

2.66008 

425.32 

555.2 

552.8 

2.4 

The  numbers  in  the  second  column  are  taken  directly 
from  Table  III.  for  the  values  of  f^  given  in  column  i.  Sub- 
tracting the  numbers  in  column  2  from  {(p)  (=0.60799)  gives 

2 
those  in  column  3;  and  these  multiplied  by  -^  {=  1. 12323) 

are    written    in   column  4.       Adding    I  (U)  (=0.93354)   to 
these  last  gives  the  values  of  /  (ti)  in  column  5. 

The  values  of  u  are  then  taken  from  Table  I.,  and  these 
multiplied  by  sec  '^  give  the  velocities  sought.  For  com- 
parison the  velocities  computed  by  Bashforth,  by  his  method 
already  explained,  are  also  given  ;  and  it  will  be  seen  that 


68  EXTERIOR    BALLISTICS. 

the  differences  between  his  velocities  and  those  computed 
by  (33)  are  practically  nil. 

This  method  of  determining  velocities  may  be  used 
without  material  error  when  the  initial  velocity  is  as  great 
as  1000  f.  s. 

Example. — The  8-inch  howitzer  is  fired  with  a  quadrant 
elevation  of  23°;  muzzle  velocity,  920  f.  s. ;  weight  of  shell, 
180  lbs.;  diameter,  8  inches.  What  will  be  the  velocity  in 
the  descending  branch  when  />  =  —  27°  54'  ?  (See  Mac- 
kinlay's  ''  Text-Book,"  page  109.) 

Here 

F=920,     Z7=  920  cos  23°=  846.86 

/(/7)  =0.40884;     log  ^  =  9.85 194 

The  computation  is  as  follows: 

(23°)  =       0.43690 
(-27°  540=— 0.55327 

log  0.99017  =  9.99571 

C 
log -  =  9.85194 

log  0.70412  =  9.84765 
I{U)=       0.40884 

I  {11)  :=  I.I  1296       .  •  .  ^27.  5^.  =  609.4  f.  S. 

Mackinlay  gets  by  Niven's  method,  dividing  the  tra- 
jectory into  two  parts,  6^270  54' =  610.6  f.  s.  It  will  be  seen 
that  by  the  method  developed  above  for  calculating  veloci- 
ties, the  length  of  the  arc  taken  makes  no  difference  in  the 
accuracy  of  the  results. 

Siacci's  Method. — Equation  (13)  may  be  written 

/^     dd-     _gC    r  ^  sec'  &  du 

Since  ^  is  a  function  of  u,  there  must  be  some  constant 
mean  value  of  sec  d-  which  will  satisfy  the  above  definite 


EXTERIOR   BALLISTICS.  69 

integral.     Representing  this  mean  value  of  sec  d-  by  a,  and 
writing  U'  and  u'  for  af/and  au  respectively,  we  have 


n     d&      _agC_  f^  _duf__ 


Making 


^(«')  =  ^7^  +  e 


(34)  becomes 

tan  ^  -  tan  ,!'  =  ^{ /(«')- 7(^7)}  (35) 

The  values  of  I  {ii')  are  given  in  Table  1.  for  oblong  pro- 
jectiles, and  in  Table  II.  for  spherical  projectiles.  The 
method  of  computing  the  /-function  is  entirely  similar  to 
that  already  described  for  the  5  and  /-functions,  and  need 
not  be  repeated.  For  oblong  projectiles  the  formulae  areas 
follows,  in  which,  for  uniformity,  /  (z^)  is  employed  as  the 
general  functional  symbol: 

2800  f.  s.  >  7^  >  1330  f.  s. : 

/  W  =  [5.3547876]  ^  —  0.028872 

1 330  f.  s.  >  7/  >  II 20  f.  s. : 
/(t/)  =[8.2947896]  ^  +  0.015293 

1120  f.  s.  >  T^  >  990  f.  s.: 
7(7')  =  [17.1436868]  -^  +0.085087 

990  f.  s.  >  z/  >  790  f.  s. : 
7(2;)  =  [8.4557375]  -L  — 0.061373 

790  f.  s.  >  z^  >  o  : 
I{v)^  [5.7369333]  -^  —  0.356474 


70 


EXTERIOR  BALLISTICS. 


If  we  compare  (34)  with  (31)  it  will  be  seen  that 

„_  i  (f).-w«  I  ^ 

( tan  (p  —  tan  §  ) 
and  this  value  of  a  renders  (34)  and  (35)  exact  equations;  in 
fact,  reduces  them  to  (31).  It  would  seem  at  first  as  if 
nothing  had  been  gained  by  introducing  a  into  (35),  since 
its  value  depends  upon  that  of  ^2,  and  must,  therefore,  change 
when  n  changes.  The  following  table  gives  the  values  of  a 
for  the  arcs  contained  in  the  first  column,  when  ;/  =  2,  w  =  3, 
and  n=z6,  computed  by  the  above  formula  : 


Arc 
Mo* 


30°  to      20° 


30^ 


10^ 


ic- 


30°  ''  —20^ 


30^ 


30^ 


I . 1066 

1-0741 
I. 0531 
I. 0419 

I .0409 

I. 0531 


I . 1069 

1.0749 
I. 0541 

I .0429 
I .0418 

I. 0541 


1079 

0772 

0573 

0460 

0443 
0573 


It  is  evident  from  this  table  that  when  the  angle  of  pro- 
jection is  as  great  as  30°,  the  velocity  at  any  point  of  the  tra- 
jectory may  be  computed  with  sufficient  accuracy  by  using 
either  set  of  values  «;  since  the  greatest  difference  between 
those  in  the  second  and  fourth  columns  on  the  same  line  is 
but  0.0042,  and  this  would  make  but  a  slight  difference  in 
the  values  of  U'  or  u'.  MoreoVer,"since  U'  —  a  Fcos  ^,  and 
u'  =^  av  cos  ^?,  it  is  apparent  that  U'  and  u'  differ  less  from  V 
and  V  respectively  than  do  U  and  u;  and  this  is  important 
when,  as  is  usually  the  case,  the  law  of  resistance  is  different 
for  the  initial  and  terminal  velocities. 

If  in  the  above  expression  for  a  we  make  n  =  2,  we  have 
Didion's  expression  for  «,  viz. : 

^^    (y)-W 

tan  (p  —  tan  ^ 


EXTERIOR   BALLISTICS.  7 1 

in  which 

(i?)  —  i  I  tan  ^  sec  &  +  log  tan  ^-  +  — )  I 

Example. — 'A  12-inch  service  projectile,  weighing  800  lbs., 
is  fired  at  an  angle  of  projection  of  30°  and  a  muzzle  velocity 
of  1886  f.  s.  Required  its  velocity  when  (a)  the  inclination 
of  the  trajectory  is  15°,  and  (b)  when  the  inclination  is —  15°. 

Here^=i:  12,  w  —  800,  V  =  1886,  and  ip  =  30°.  From  (35) 
we  get 

/  {u')  =:  /(^')  -f  -^  I  tan  ^  -  tan  ??  i 
(a)  ^  =  15°.     From  our  data  we  have 

„^      (30°)-(.5°)   ^^gd3g821^,.o888 
tan  30°  —  tan  15°       0.30940 

U'  =  a  Fcos30°  =  1778.34     .'./{[/')  =  0.04270 

^  _  w 800 

d^       144 
and 

tan  30°  —  tan  15°  =0.30940 

Tt>\  X      288    X    0.30940 

.-./(.)=  0.04270  +    3^  ^  ^3^3^  -  0.14500 

.-.  «'=  1149.77. 

1149.77  . 

.  • .  z/,.o  ==  — ^^-^  =  1093.3  I-  s. 
"        a  cos  15°  ^^  ^ 

(b)  x^=  —  15°.     We  have 

^^      (30°) +  (15°)       ^0.87911  ^  J  ^.^ 
tan  30°  +  tan  15°       0.84530  "^ 

C/'  — «  Fcos  30°=  1698.65     .-.  7(^0  =  0.04958 
tan  30°  +  tan  1 5"  ==  0.84530 
.-.  I {u')  =0.04958  +  0.29260  =  0.34218 
.-.  ?/'  =  891.14 
.-.  z;,,,.  =887.1  f.  s. 

The  values  of  v^^^  and  z/_,^,  computed  by  (31)  are  1097.6 
and  892.9  respectively. 


72  EXTERIOR   BALLISTICS. 

Siacci's  Modification  of  (35)  for  Direct  Fire.— 

Since  in  direct  fire  the  angle  of  projection  does  not  exceed 
15°,  and  is  generally  much  less,  the  values  o^  a  for  this  kind 
of  fire  will  not  differ  much  from  unity.  For  example,  with 
10°  elevation,  and  an  angle  of  fall  of  —  12°,  we  shall  have 
for  a 

,,_       (10°) +  (12°)       _Q.39i39_^^^., 
■"  tan  10°  +  tan  12°  ~  0.38889  "~  "^ 

It  is  manifest,  therefore,  that  for  sucli  small  angles  no 
material  error  would  result  in  making  «=  i ;  the  following, 
however,  is  a  closer  approximation.  If  we  consider  that 
part  of  the  trajectory  lying  above  the  horizontal  plane 
passing  through  the  muzzle  of  the  gun,  it  will  be  seen  that  * 
a  should  be  greater  than  unity  and  less  than  sec  co.  Siacci 
makes 

W-2 

a  =  (sec^)«-i 


therefore,  when  «  =  2,  a  =1 ;  when  «  =  3,  a  =  V  sec  ^,  and 

when  n  =^  6^  a  =z  sec    f  ;  and  the  average  value  of  a  for  the 
whole  trajectory  generally  fulfils  the  above  condition. 

This  value  of    a  substituted   in   (34)  gives,  by  an  easy 
reduction, 

•tan  c^  -  tan  ^  =:      /^  ,     \  . L-—  -  -L  \ 

- —  ^  n  A  cos  (p  {  {u  sec  cpf       F"  ) 

or,  writing  u'  for  u  sec  ^,  and   proceeding  as  already  ex- 
plained, 

'      Example. — Take  the  follow^ing  data: 

800 
^=  12  ;  7£/  =  800;  6'= ;    F=  1886  ;  ^  =1  10°.     Compute 

144 
the  remaining  velocity  in  the  descending  branch  when 
t?=^  13°.     We  have 

/  {u')  =  -^  cos"  (f  (tan  ^  -  tan  ^)  +  /  (F) 


EXTERIOR   BALLISTICS. 

and  the  computation  will  be  as  follows: 

log  (tan  10°  +  tan  13°)  =  9.60980 

log- -^  =  9.55630 

2  log-  cos  10°  =  9.98670 

log  0.142 1 7  =  9.15280 
7(1886)1=0.03477 

/  (//')  z=  0. 1 7694  //  =  1 07 1 . 76 

IO7T.76  COS  lo"^ 


n 


COS  13° 


=  1083.2  f.  s. 


The  velocity  at  the  same  point  computed  by  (31),  divid- 
ing- the  trajectory  into  three  arcs,  with  the  points  of  division 
corresponding-  to  velocities  of  1330  f.  s.  and  1120  f.  s.  respec- 
tively, is  7' =:  1081.55  f.  s.  This  agreement  is  very  close; 
but  if  we  make  if  =  30°  and  ^  ==  15°,  as  in  the  preceding  ex- 
ample, we  should  find  by  tiiis  method  ?''i5«  =  1113.1;  and  if 
d- =z  —  15°,  we  should  find  7^_i^«  =  859.3,  which  differ  consid- 
erably from  their  true  values. 

Mven's  Method.— W.  D.  Niven,  Esq.,  M.A.,  F.R.S., 
has  given  the  following  method  for  determining  velocities 
in  terms  of  the  inclination  : 

Equation  (13)  may  be  written 

J  ,  ^  Aj  ,    («sec.?r"        "-A J  „.   «'"*■ 

in  which,  as  before,  a  is  some  mean  value  of  sec  d^  between 
the  limits  sec  <p  and  sec  ??,  and  1/  =z  av  cos  d^  and  U'  ^=.a  Fcos  (p. 
Integrating,  we  have 

'  an  A    \  //«        'U'-S    ~  a    InA   u'^       nA    U'^S    ^^^^ 

iplying  botl" 
degreed,  and  making 


Multiplying  both   members   by  - —  to   reduce  ^  —  ?^  to 


i^(,-^)  =  Z) 


74  EXTERIOR   BALLISTICS, 

and 


n  TT  A    n""        ^         ^ 
the  above  equation  becomes 

n  =  ^\D{,/)-j?{u')\*  (38) 

which  is  the  equivalent  of  Niven's  expression  for  the  velo- 
city and  inclination.  Mr.  Niven  has  published  a  table  of  the 
/>>•  function  for  velocities  extending-  from  400  f.  s.  to  2500  f.  s. 
(See  Table  VI.  ii  Mackinlay's  "Text-Book.")  It  will  be 
seen  by  comparing  the  expressions  for  D  {v)  and  I  [v)  that 
we  have  the  relation 

and,  therefore,  in  terms  of  the  /-function,  (38)  becomes 

.^  =  t|'{^M-/(^0}  (39) 

log  ^==1.4570926 

Comparing  (37)  with  (31),  it  is  apparent  that  to  make  (37) 
or  (38)  exact  equations  we  must  have 


-\^^r 


For  direct  fire  Didion's  value  of  a  may  be  used ;  but  for 
high-angle  firing-  the  following  gives  more  accurate  results, 
obtained  from  the  above  equation  by  making  a/  =  2  : 


a  = 


i'^f[ 


Example. — Take  the  following  data: 
df=l2;     w=r8oo;      F=:  1886;     ^  —  30°   and    <>  =  —  30°  ; 

/?  =  30°  +  30°  =  60°  ;  to  find  v,^.. 

% 

*  If  we  use  Niven's  tables,  in  which  the  functions  decrease  with  the  velocity.  (38)  should  be 
written 

i>«£j/?(i/0-^(«')[ 


EXTERIOR   BALLISTICS. 


75 


We  have  from  (38) 

D{u')  =  D{U')  +  ^D 

The  computation  may  be  conveniently  arranged  as  fol- 
lows : 

log  (ip)  =  978390 

constant  =  1.758 12 

c  log  30  =  8.52288 


log 


3)0.06490 

log  a  =  0.02163 

log  D^  177815 

r  log  (7=  9.25527 

11.3516=  1.05505 


log  F=  3.27554 

log  a  =  0.02163 

log  cos  ip  =  9.93753 

log  U'  =  3.23470 
6^'=  1716.74 


(Niven's  Table)      D  {U')  =  84.6090 


C 

D{u') 


Dzzz  II. 3516 


73-2574 


.*.«'  =  827.12  =:«  Z/  cos  ?? 

.  • .  v.j^^.  =  908.7  f.  S. 

Siacci's    method,    using    Table    I.    of    this    work,   gives 

^_3oo  =  907.5  f.  s. ;    while  equation  (31)  gives  v_^^  =  913.2  f.  s. 

Modificatioii  of  (38)  for  Direct  Fire. — If  we  make 

a  =  (sec  (fY^ 

we  shall  have,  by  a  process  similar  to  that  already  employed 
in  Siacci's  method,  the  following  modified  form  of  (38), 
which  can  be  used  in  all  problems  of  direct  fire,  viz.: 

C 


j9  = 


cos  (f 

in  which  u'  ^=^u  sec  ip. 
Example, — Let  <a^  =  1 2  ; 


\^D  iu')  -  D  {y)\^ 


w 


800;       V: 


(40) 

\ 

886;     ^  =  10°; 


=  —  13®.     The  computation  is  as  follows : 


y6  EXTERIOR   BALLISTICS. 

log  Z^=  1. 36173 

log  cos  (f  =z  9.99335 

c\oo;  C=  9-25527 
log  4.0771  =  0.6T035 
i;  (1886)  =  84.9966  ^^^^ 

D  («0  =  80.9195     .-.//=  1068. 14  =  z'-^;^^ 
.' .  V  =  1079.6  f.  s. 

which  is  within  2  feet  of  the  value  of  z/  computed  by  the  exact 
formula  (31).  This  modified  form  of  Niven's  method,  for  sim- 
plicity and  accuracy,  seems  to  leave  nothing  to  be  desired. 

For  small  angles  of  projection,  say  not  exceeding  5°,  we 
may  put  ?/  =  v,  and  cos'^  =  i ;  and  (40)  becomes 

%  Example, — In  the  preceding  example  suppose  ^  =  3°. 
What  will  be  the  value  of  d-  when  the  velocity  is  reduced 
to  1500  f.  s.? 

(a)  By  Niven's  Table  : 

Z>  (1886)  =  84.9966 
Z>  (1500)==  83^9359, 

log     1.0607  =:  0.02560 

log  6^  =  0.74473 

log  D  =  0.77033 

/>=5°.89  =  3°-'> 
.-.  ??=  -  2°.89 

(b)  By  Table  1. : 

7(1500)=  0.07173 
/(i886)  =  0.03477 


log  0.03696  =  8.56773 

log?=  145709 
log  (7  =  0.74473 

log  7^  =  0.76955 
D  =  5°.88 
.-.  ^=  -2°.88 


CHAPTER  VI. 

TRAJECTORIES— HIGH-ANGLE   FIRE. 

As  we  have  seen,  the  differential  equations  for  x,y^  t,  and 
s  do  not  generally  admit  of  integration  in  finite  terms  for 
any  law  of  resistance  pertaining  to  our  atmosphere ;  that 
is,  for  any  recognized  value  of  ?i.  It  is  true  that  Professor 
Greenhill  has  recently*  succeeded,  by  a  profound  analysis, 
in  deducing  exact  finite  expressions  for  x  and  y  by  means  of 
elliptic  functions,  when  ?^  =  3 ;  but  these  results,  though  of 
great  interest  to  the  mathematician,  are  far  too  complicated 
for  the  practical  use  of  the  artillerist.  When  /^  =  2  the  ex- 
pression for  ds  can  be  integrated  and  useful  results  deduced 
therefrom,  as  will  be  seen  further  on. 

For  low  velocities,  such  as  are  generally  employed  in 
high-angle  and  curved  fire,  the  effect  of  the  resistance  of 
the  air  upon  heavy  projectiles  is  comparatively  slight;  and 
for  a  first  (though  rough)  approximation  we  may,  in  such 
cases,  omit  the  resistance  altogether,  or,  better  still,  we  may 
suppose  the  projectile  subject  to  a  mean  constarit  resistance. 
A  still  closer  approximation  may  be  obtained  by  taking  a 
resistance  proportional  to  the  first  power  of  the  velocity. 
As  the  differential  equations  for  the  co-ordinates  and  time 
are  susceptible  of  exact  integration  upon  each  one  of  these 
hypotheses,  we  will  consider  them  in  turn. 

TRAJECTORY   IN  VACUO. 
Making  p  =  o,  (9)  becomes 

duzuzo 
and  therefore,  in  vacuo,  the  horizontal  velocity  is  constant,  or 

/^=  U 
Integrating  (21),  (22),  (23),  and  (24)  between  the  limits 
(p  and  d-  gives,  \i  ti  ^  U, 

*  "  Proceedings  of  the  Roj^al  Artillery  Institution,"  Vol.  XI. 
10 


78  EXTERIOR   BALLISTICS. 


and 


/  =  —(tan  f  —  tan  d)  (4O 

;ir  =  —  (tan  ip  —  tan  d)  (42) 

772 

7  =  — '(tan'  ip  ~  tan'  />)  (43) 


((^)  -  ('>))  (44) 


Equation    of    Trajectory  in    Vacuo. — Eliminating 
tan  d-  from  (42)  and  (43)  gives 

y  ^=L  X  tan  (p 


2W 

which  is  the  equation  of  a  parabola  whose  axis  is  vertical 
A  parabola,  therefore,  is  the  curve  a  projectile  would  de- 
scribe in  vacuo. 

Since  a  parabola  is  symmetrical  with  respect  to  its  axis, 
the  ascending  branch  is  similar  in  every  respect  to  the  de- 
scending branch,  the  angle  of  fall  being  equal  to  the  angle 
of  projection  ;  and  generally,  for  the  same  value  of  j,  tan  d^ 
has  numerically  the  same  value,  but  with  contrary  signs,  in 
both  branches;  being  positive  in  the  ascending  branch, 
negative  in  the  descending  branch,  and  zero  at  the  vertex. 

If  we  make  ?^  =  —  ^  in  (42)  it  becomes 

^-       2  C/'  V  sin  2ip 

X  = tan  <p  = i 

g  g 

and  this,  for  a  given  velocity,  is  evidently  a  maximum  when 

f  =  45°. 

Subtracting  (42)  from  the  above  equation,  and  reducing, 
gives 

X  —  X—  — (tan  <p  +  tan  ^) 

2  tan  ^  ^       ^    '  ^ 

also,  dividing  (43)  by  (42)  gives 

f  =  ^(tan^  +  tane?) 

whence 

:^  =  ^(-^-^)tan^  (45) 


EXTERIOR   BALLISTICS.  79 


Making  ??  =  —  ^  in  (41),  we  have 

^       2U ^  2V    . 

Y  rzz tan  w  =  —  sin  <p 

g  g 

Subtracting  (41)  from  this  last  equation  gives 

T  —  t=—  (tan  (p  +  tan  d) 
also,  (43)  divided  by  (41)  gives 

-7  =  7  (tan  ^  + tan??) 


^{T-t)  f46) 


whence 

2 
Dividing  (44)  by  (42)  gives 

s^    (y)-(^)    ^^ 

X       tan  (f  —  tan  d^ 

Didion's  «,  then,  is  the  ratio  of  a  parabolic  arc  whose 
extremities  have  the  same  inclination  as  the  arc  of  the  tra- 
jectory under  consideration,  to  its  horizontal  projection. 

Expression  for  the  Velocity. — From  (43)  we  have, 
since  V  cos  (p  zz^v  cos  d-  =  U, 

v"  sin''  d-z^zV  sm^  (p  —  2gy. 

Adding  v"  cos''  d^  to  the  first  member,  and  its  equal, 
V  cos"*  (f,  to  the  second  member,  and  reducing,  we  have 

v''  =  V  -  2gy 

If  h  is  the  vertical  height  through  which  the  projectile 
must  fall  to  acquire  the  velocity  of  projection  (F),  we  shall 
have  Tza  7 

V^  zzz2  gk 

and  this  substituted  in  the  above  formula  gives 

v''  =  2g{h-y) 
that  is,  the  velocity  of  the  projectile  at  any  point  of  the 
trajectory  is  that  which  it  would  acquire  by  falling  through 
a  vertical  distance  equal  to  ^  —  j^. 

All  the  properties  of  the  trajectory  in  vacuo  may  be 
easily  and  elegantly  determined  by  means  of  the  funda- 
mental equations  (41)  to  (44)  inclusive. 


80  EXTERIOR   BALLISTICS. 

CONSTANT  RESISTANCE. 


P    _ 


Suppose  the  resistance  constant,  and  put  ~  ^  m  ;    then 

zv 

the  elimination  of  dt  from  (9)  and  (12)  gives 

du  d& 

in 


u  cos  ?> 

whence  ^ 

log  u  =  m  log  tan  1  -  -] I  -f-  ^. 

Let  v^  be  the  velocity  when  ?>  =  o,  that  is,  at  the  summit 
of  the  trajectory  ;  then  C  —  log  v^,  and  we  have 


(f+4)  <«) 


r=  2/„  tan 

4 

Substituting  this  value  oi  u  in  equations  (21)  to  (24),  and 

integrating   so  that  /,  x,  y,  and  s  shall  all  be  zero  at  the 

origin,  that  is,  when  -&  =npy  we  have,  making  the  necessary 

reductions, 

TT  sm  w  —  m  sin  i^  —  m 

j^2  COS  (p  (sin  ^  —  2m)         ^  cos  ?^  (sin  d  —  2m) 

~  g{i  —  4^')        ""  ^        <^  (i  -  4^^') 

j^2  I  +  sin  ^  (sin  (p  —  2m)       ^9 1  +  sin  d-  (sin  ?^  —  2/;?) 

^  ~  4^(1  -Iff)  ""  4^(1  -f^) 

,j,  cos^  (f  -{-2m  (sin  ip  —  in)       ^  ^  cos''  d-  ~\-2m  (sin  z?  —  ??/) 

4w^(i  —  ^^z'')  4^«^(i  —  ^^^) 

When  2?;^  =  i,  the  differential  expression  for  x  becomes 
logarithmic,  as  do  those  for  /,  y,  and  s  when  in  —  \.  The 
integrations  are  easily  obtained  for  these  values  of  m^  but 
are  omitted  on  account  of  their  length,  and  as  being  of  no 
great  practical  importance.  In  the  application  of  these  for- 
mulae it  will  be  necessary,  since  the  resistance  of  the  air  is 
not  constant,  but  varies  with  the  velocity,  to  determine  a 
proper  mean  value  for  m  between  the  limits  of  integration  ; 
and  this  we  may  do  as  follows :  After  having  computed  the 
horizontal  velocities  u^  and  u^  by  means  of  (33),  corre- 
sponding to  the  inclinations  a  and  /9,  the  value  of  in  may  be 
determined  by  the  following  equation  deduced  from  the 
above  expression  for  71 : 


EXTERIOR   BALLISTICS.  8 1 

^jj^  ^ log  u^  -  log  U^ 

logta„(^  +  f)-logta„(^+4) 

Example. — Compute  the  values  of  t,  .r,  y,  and  s,  from 
(p  =  30°  to  ??  =  o,  with  the  data  given  on  pag-e  6y.     We  have 

,,,  _  log  75 1  +  log  cos  30"  -  log  525.91    _^^.^_ 
'"^  -  log  tan  60°  -  ^-^^^^^ 

Substituting  in  the  above  formulae,  we  find 

^  =  3-I073  +  7.4295  =  10^537 
X  =  16908  —  10557  =  6351  ft. 
y  =    4446  —    2526  =  1920  ft. 

^=:   III55    —     4578    =   6577  ft. 

Bashforth  gets,  by  dividing  the  arc  into  8  parts, 
t  =  io''.4i3,  X  =  6074  ft.,  and  7  =  1882  ft. 

It  is  easy  to  see  how  by  suitable  tables,  the  construction 
of  which  offers  no  difficulty,  the  time  and  co-ordinates  ma}^ 
by  this  method  be  readily,  and  for  arcs  of  limited  extent 
accurately,  computed.     For  example,  we  have 

x  =  A  V~A'  v' 
A  being  a  function  of  m  and  ^,  and  A^  the  same  function  of 
m  and  d^. 

RESISTANCE     PROPORTIONAL    TO   THE    FIRST   POWER   OF  THE 

VELOCITY. 

Differential  Equations. — When  ;/  =  i,  the  differential 
equations  (13),  (17),  (18),  and  (19)  become  respectively,  since 

tiA 


CO^  ^^ 

dt=- 

k  du 
g    u 

dx  = 

k 

du 

dy  =  - 

k 

tan  -&  du 

82  EXTERIOR   BALLISTICS. 

Time  and  Co-ordinates. — The  integration  of  the  first 
three  of  these  equations  between  the  limits  {cp,  d)  and  {[/,  7i) 
gives  (supposing  k  constant) 

tan  <p  —  tzin^  =  k(~  —  ^')  {48) 

or,  using  common  logarithms, 

f=M-\og-  (49) 

in  which  M  =  2.30259;  and 

:^=:^{U-u)  (50) 

Substituting  for  tan  ?^  in  the  expression  for  dj/  its  value 
from  (48),  it  becomes 

dy= (7-,  -^  tsiu  <p)  du -\ 

or 

dj/=  IjY  -{-  tan  ^  j<^4r  —  /&^^ 

whence,  supposing  y  to  vanish  with  x  and  if, 

7  =  (-^  +  tan  <pj  x  —  kt  (51) 

Determination  of  h. — In  the  above  integrations  we 
have  assumed  k  to  be  constant,  whereas  it  varies  with  the 
velocity  ;  but  our  results  will  be  correct  if  we  give  to  ^  a 
proper  mean  of  all  its  values  between  the  limits  of  integra- 
tion ;  and  as  k  varies  slowly  and  with  considerable  regularity 
for  all  velocities  for  which  this  method  will  be  used,  we  will 
take  for  k  the  value  corresponding  to  the  arithmetical  mean 
of  the  two  velocities  at  the  extremities  of  the  arc  under 
consideration.  It  is  evident  that  the  smaller  the  arc  of  the 
trajectory  over  which  we  integrate,  the  less  will  be  the 
error  committed  in  taking  this  value  for  k.      But  it  will  be 


EXTERIOR   BALLISTICS. 


83 


shown  by  examples  that  no  material  error  will  result  for 
velocities  less  than  about  1000  f.  s.,  when  the  whole  tra- 
jectory is  divided  into  two  arcs  with  the  point  of  division  at 
the  summit. 

When  ;/  =  I,  we^have 


w 


whence  from  (6)  and  (7) 


C 


(loooy 


C  m        (say) 


The  following  table  gives  the  values  of  tn  for  velocities 
extending  from  900  f.  s.  to  500  f.  s.,  with  first  differences  : 


TABLE   OF   7n. 


V 

m 

d, 

V 

tn 

d, 

500 

32.814 

66^ 

710 

23.700 

346 

510 

32.146 

618 

720 

23.354 

357 

520 

31.528 

572 

730 

22.997 

340 

530 

30.956 

554 

740 

22.657 

323 

540 

30.402 

539 

750 

22.334 

335 

550 

29.863 

527 

760 

21.999 

376 

560 

29.336 

490 

770 

21.623 

388 

570 

28 . 846 

427 

780 

21.235 

372 

580 

28.419 

392 

790 

20.863 

358 

590 

28.027 

387 

800 

20.505 

344 

600 

27.640 

384 

810 

20.161 

384 

610 

27.256 

381 

820 

19.777 

448 

620 

26.875 

382 

830 

19.329 

433 

630 

26.493 

382 

840 

18.896 

442 

640 

26.  I  II 

356 

850 

18.454 

426 

650 

25-755 

388 

860 

18.028 

412 

660 

25.367 

365 

870 

17.616 

398 

670 

25.002 

343 

880 

17.218 

385 

680 

24.659 

321 

890 

16.833 

372 

690 

24.338 

300 

900 

16.461 

359 

700 

24.038 

338 

84 


EXTERIOR   BALLISTICS. 


The  value  of  k  in  the  ascending-  branch  will  be  assumed 
to  be  that  due  to  the  velocity  |  {y-\-v^\  and  in  the  descend- 
ing branch,  to  \  (^o  +  ^e)*  ^e  being  the  velocity  at  the  point  of 
fall.  The  first  step,  then,  is  to  compute  v^  and  Vq  ;  and  this 
can  readily  be  done  by  means  of  (33),  as  already  explained. 

Expressions  for  the  Ascending  and  Descending- 
Branches. — It  will  be  seen  that  x,  y,  and  t  are  functions  of 
6^  and  u;  and  these  latter  depend  upon  (p  and  ?^,  as  shown  in 
equation  (48). 

From  this  equation  we  have 


■^  +  tan  ^ 


-  +tan?^  =  - 


in  which  u^  is  the  value  of  21  at  the  summit;  whence 

k 


U 


-f-  tan  <p 


and,  since  d-  is  negative  in  the  descending  branch, 

k 


^fl  = 


+  tan  d- 


(52) 


(53) 


The  following  expressions  for  t,  x,  and  y  for  the  ascend- 
ing and  descending  branches  are  easily  deduced  from  (49), 
(50),  and  (51),  in  connection  with  (52)  and  (53): 


ASCENDING  BRANCH. 

'0  =  ^— log  — 


=}{--) 


DESCENDING  BRANCH. 


k 
JJ^o  =  ■—  ^0 


kL 


ye 


(^o— ^e) 


kto 


In  using  these  formulae,  u^  and  Uq  are  to  be  computed  by 
means  of  (52)  and  (53). 

The  zero  subscript  is  to  be  interpreted  "  from  the  origin 
to  the  summit";  and  the  theta  subscript  "from  the  summit 


EXTERIOR   BALLISTICS.  85 

to  a  point  in  the  descending  branch  where  the  inclination 
is^^/' 

The  method  of  computing  a  trajectory  by  these  simple 
formulas  will  be  best  exhibited  by  examples,  which  we  will 
select  from  those  that  have  been  worked  out  by  other 
methods  of  recognized  accuracy,  or  which  have  been  tested 
by  firing. 

Example  i. — -Calculate  the  trajectory  with  the  data  on 
page  6^^  viz.  : 

F=75if.  s. ;   ^  =  30°  (whence  C/'=  Fcos^  =  650.385);  d:=. 

2  2(1'' 

6.27  inches;  w=.jo  lbs.  (whence -^  = =  1.12323). 

Assuming  —  37°  to  be  the  angle  of  fall,  we  will  divide 
the  trajectory  into  two  arcs,  the  first  extending  from  30°  to 
0°,  and  the  second  from  0°  to  —37°.  The  velocities  v,,  and 
z/_370  are  computed  as  follows : 

From  Table  III.  we  take  out  (30°)  =  0.60799,  and  (37°)  = 
0.81977;  and  from  Table  I.,  /(^)  =  / (650.385)  =  0.93354. 
Then 

2 
—  (30°)  =  1. 1 2323  X  0.60799  =  0.68291 

/(^):zz  0.93354 

I{v^=  1.61645 
(Table  I.)        v^  =  525.91 
2 
-^(37°)  =  1-12323  X  0.81977  =  0.92079 

I  (y^  =  1.61645 

/(^^_3,o)  =  2.53724 

z/_3,«  =  434.25  sec  37°  ==  543.74  f.  s. 

The   mean  velocity  from  which  to  determine  k  in  the 
ascending  branch  is  i  (751  +  525.91)  =  638  f.  s. ;    whence 
m  =  26.187.      The  remaining   calculations   may   be   conve- 
niently arranged  as  follows: 
II 


86  EXTERIOR    BALLISTICS. 


log  m  =  1.4180857 

log  (::  =  0.2505630 

log  ^=  1.5077210    U=  32.19) 

log  k  =  3-1763697 
log  C/=  2.8131705 


log  2.3078  =  0.3631992  =  log  jj 

[Equation  (52)]  tan  <p  =  0.5774 

log  2.8852  =  0.4601759  (sub.  from  log  X^ 

log  ?^o  =  2.7161938 
u^  =  520.228 

^=r  650.385 

log  130.157  =  2.1144675 
log -  =  1.6686487 

s • 

log;ro  =  3.783 1162 
x^  =  6069  ft. 
Bashforth  gets  by  8  steps,  6074 

Difference,  5  ft. 

log  17=  2.8 1 3 1705 
log  «;==  2.7161938 

log  0.0969767  =  8.9866674 

log  J/=:  0.3622157  (add  log-— j 

.    .  log /o  =1.0175318 

Bashforth'gets       10^413 

Difference,  o''.ooi 

log  — ^  =  1.0669224     (add  log  k) 

4.2432921  =  log  175 10 
log  kt\  —  4.1939015  —  log  15628 

y,^  1882 

Bashforth  gets  1882 

Difference,  o 


EXTERIOR   BALLISTICS.  87 

These  results,  being  practically  identical  with  those  de- 
duced with  vastly  greater  labor  by  Prof.  Bashforth,  pnn^e 
that  when  the  law  of  resistance  is  that  of  the  square  of  the 
velocity,  as  in  this  example,  we  may  get  quite  as  close  an 
approximation  to  the  true  trajectory  by  assuming  that  the 
resistance  is  proportional  to  the  first  power  of  the  velocity 
as  we  can  upon  the  hypothesis  of  the  law  of  the  cube,  and 
with  a  great  gain  in  simplicity  and  labor. 

We  have  next  to  compute  the  descending  branch  from 
f^  =0°  to  3  =  —  37°.  The  mean  velocity  from  which  to 
determine  k  in  this  branch  is 

i  (525.91  +  543.74)  =  534.8  f.  s. 
whence  m  =  30.690. 

log  m  =  1.4869969 
log  6' =  0.2505630 
log  £  =  1. 5077210 

log  k  =  3.2452809 
log  z^o  =  2.7209 114 

k 
[Equation  (53)]    log  3.34480  =  0.5243695  =  log  — 

tan  37°  =  0.75355 


log  4.09835  =  0.6126090 


log  2^.3,0  =  2.6326719 

f^_3,o  =  429.21 
^0=525.91 


log  96.70    =  1.9854265 

log  7=  1-7375599 

log  ^-„«  =  3.7229864 

^_3,0=  5284    ft. 


88  EXTERIOR   BALLISTICS. 

log  2^0  =  2. 7209 114 
log  ?/_3,o  =  2.6326719 

log  0.0882395  =  8.945663 1 
log  ^]/  =  0.3622157 

log/-3,o=  1.0454387 


/_3,o3:zIlM03 
k 

log— ^.3,0  =  4.2473  5  59  =  log  17675 
log  k  t.,,.  =  4.2907196  =  log  1953 1 


J/_3,,=    -  1856    ft.        > 

The  projectile  is  still  1882  —  1856  —  26  ft.  above  the  level 
of  the  gun  =  Ay.  If  Ax  and  At  are  the  corresponding  addi- 
tions to  the  range  and  time  of  flight,  we  shall  have  approxi- 
mately 

Ax 
Ax  =  26  cot  37°  =r  35  ft. ;  and  At  = =  o''.o8o. 

We  therefore  have 

^=6069  +  5284 -f- 35  =  11388  ft. 
r=i  10.412  -[-  II. 103  +  0.080  =  2i'^595 

These  values  agree  almost  exactly  with  those  deduced 
by  interpolation  from  the  table  on  page  117  of  Bashforth's 
work. 

Example  2. — The  8-inch  howitzer  is  fired  with  a  quad- 
rant elevation  of  23°.  Muzzle  velocity,  920  f.  s. ;  weight  of 
shell,  180  lbs.  ;  diameter,  8  inches.  Find  the  range  and 
time  of  flight.  (Mackinlay's  "Text-Book  of  Gunnery," 
page  107.) 

Assuming  the  angle  of  fall  to  be  —  27°  54',  we  find  by  the 
above  method 

X=z  7886  +  7108—  13  =  14981  ft. 

T  =  10.183  +  10.801  —  0.022  =  20^^.962 

Mackinlay  gets,  using  Niven's  method, 

X—  14787  ft,  and  r=2o".8i3 
He   states  that  "the  published  range-table  gives  15000ft. 
as  the  range,  and  2i'^5  for  the  time  of  flight." 


EXTERIOR   BALLISTICS.  89 

Example  3. — Let  V  =.  2g%  m.  =  977.71  ft.,  d ^=  \^  cm., 
w  =  io  k.g-.,  f  =  35°  21',  o  —  1.270  k.g-.,  and  0,=  1.206  kg. 
Find  Xand  T.  (Krupp's  Bulletin,  No.  55,  December,  1884.) 
For  the  Krupp  projectiles  and  low  velocities  we  will 
take  for  c  the  ratio  of  the  coefficients  of  resistance  deduced 
from  the  Krupp  and  Bashforth  experiments  respectively, 
and  which  are  given  in  Chapter  II.  Let  these  coefficients 
be  represented  by  A  and  A' .  Then  for  velocities  less  than 
790  f  s.  we  have 

10^^  =  5.6698755-  10 
log  ^'=5.7703827  —  10 
log  c  =  9.8994928 
.•.^=0.7934 
To  find  cT,  expressed  in  English  units,  when  w  and  d  are 
given  in  kilogrammes  and  centimetres  respectively,  we  have 

^  _  loooo  k  w 
~  \^m^  c  d' 
in  which  k  is  the  number  of  pounds  in  one  kilogramme,  and 
in  the  number  of  feet  in  one  metre.     Reducing,  we  have 

C-=  [1.2534887]  J 

As  the  initial  velocity  in  this  example  is  considerable, 
we  will  take  into  account  the  density  of  the  air  at  the  time 
the  shots  were  fired,  and  also  the  diminution  of  density  due 
to  the  altitude  attained  by  the  projectile;  and  for  this  pur- 
pose we  will  assume  the  mean  value  of  y  for  the  whole  tra- 
jectory to  be  2000  ft. 

The  complete  expression  for  (7  is  (Chapter  VII.), 

from  which  we  determine  log  6"  as  follows: 

log  w  =:  1.4771213 

^  log  ^^  =  7.6478175 

constant  log  =  1.2534887 

log  0,  =  0.0813473 

^  log  ^  =  9.8961963 

z 
log  eh  =1:0.0312468 


log  (7=0.3872179 


90  EXTEklOR   BALLISTICS. 

Assuming  the  angle  of  fall  to  be  —  44°  40',  and  proceed- 
ing as  in  the  first  example,  we  find 

X=  10408  +  8736  +  104  =  19248  ft. 
7^r=:  15.088  +  16.324  +  0.221  ==  3i''.633 
Krupp  gives  the    ranges  of  three  shots  fired  with  the 
initial  velocity  and  angle  of  departure  of  this  example,  and 
the  ranges  reduced  to  the  level  of  the  mortar,  as  follows: 


NO.    OF   SHOT. 

RANGE   IN   FEET. 

18 

19039 

19 

19265 

20 

19364 

Mean  of  the  three  shots  =  19223  ft. 
Computed — mean  =       25  ft. 

Example  4. — Given  F=  206.6  m.  —  677.834  ft.,  d  =  21 
cm.,  w  =  gi  k.g.,  and  (f  =  60°,  to  find  Jfand  T.  (Krupp's 
Bulletin,  No.  31,  Dec.  30,  1881.) 

It  will  be  found  that  (assuming  the  angle  of  fall  to  be 
—  63°  30',  and  taking  no  account  of  atmospheric  conditions) 

^  =  5390  +  4945  +  67  =  10402  ft. 
T=  17.016+17.543+0.250  =  34''.8o9 
Krupp  gives  the  observed  ranges  of  five  shots,  with  the 
above  data,  as  follows : 

NO.   OF   SHOT.  OBSERVED  RANGE. 

22  10332  ft. 

23  10305  " 

24  10384  " 

25  10463  " 

26  10440  ** 

Mean  of  the  five  shots  =  10385  ft. 
Computed — mean  =        17  ft. 

Example  5. — Given  F=:  204.1  m.  =  669.63  ft.,  </  =  21  cm., 
z£/  =  91  k.g.,  and  (p  —  45°,  to  find  X  and  T.  (Krupp's  Bul- 
letin, No.  31,  January  19,  1882.) 

Assuming  the  angle  of  fall  to  be  —  49°,  we  find  as  fol- 
lows: 

^=6152  +  5678  +  56  =  1 1886  ft. 

r=  13.817  +  14.238  + 0.147  =  28^202 


EXTERIOR   BALLISTICS.  9 1 

The  following  ranges  were  measured  at  Meppen : 

NO.    OF   SHOT.  OBSERVED   RANGE. 

71  11923  ft. 

72  I  1920    " 

73  11841  " 

74  1 1 808  '' 

75  11749  *' 

Mean  of  the  five  shots  =  11 848  ft. 
Computed— mean  =        38  ft. 

Example  6. — Compute  JTand  2"  with  the  data  of  the  pre- 
ceding example,  except  that  ^  —  30°. 

Assuming  the  angle  of  fall  to  be  —  33°,  we  find  as  follows  : 

X=  5478  +  5 143  +  26  =  10647  ft. 

r=  9.908  -)- 10.183  -f  0.054  =  20".  145 

Krupp  gives  as  the  mean  of  five  measured  ranges, 
Jf  =  10779  ft. 

Computed  — mean  =  —  132  ft. 

euler's  method. 

Expression  for  s. — If  we  make  nz=z2,  that  is,  suppose 
the  resistance  of  the  air  proportional  to  the  square  of  the 
velocity,  we  shall  have  from  (20) 

C  du 

^  —  ~  'aH. 
whence,  integrating  and  supposing  j  =  o  when  u  =z  U,   we 
have 

therefore  (page  52) 

s  =  CiS{u)-S{U)^  (54) 

which  gives  the  length  of  any  arc  of  a  trajectory  when  the 
resistance  is  proportional  to  the  sqiiare  of  the  velocity,  by 
means  of  the  table  of  space  functions. 

We  may  also  obtain  another  expression  for  s,  better 
suited  to  our  purpose,  as  follows: 


92  EXTERIOR    BALLISTICS. 

Since 


'      J  c 


COS' 

we  have,  when  n=^2, 

d(d)  =  -^^  =  sec  ?^  ^  tan  & 

^    ^         cos    ?7 

and  this  substituted  in  (28)  gives 

in  which 

(?^)  =  i  I  tan?? sec ^?  + log  tan  ^-  +  ^^  I 

whence,  integrating  between  the  limits  ip  and  ??,  we  have 

or,  if  we  use  common  logarithms, 

in  which  J/=  2.30259. 

Expressions  for  a?  and  t/.— Equation  (55)  gives  the 
value  of  s  from  the  origin.  If  /  is  the  length  of  an  arc  of 
the  trajectory  from  the  origin  to  where  the  inclination  is  d-\ 
and  s"  the  length  to  some  other  point  further  on  where  the 
inclination  is  d-"  (??'>  W),  we  shall  have  from  (55) 

/=i:^— log      ^        ^      - 

and 

whence 

s"  —  s'  —  As—'M  —  los- 

If??''  differs  but  little  from  ??'  (say  one  degree),  the  cor- 
responding values  of  Ax  and  Ay  can  be  calculated  with  sufli- 


«■ 

-(f) 

«- 

-(*") 

w- 

-(f) 

(0- 

-  (^") 

EXTERIOR   BALLISTICS.  93 

cient  accuracy  by  multiplying  Js  by  cos  ^  {d-'  -\-  d-")  for  the 
former,  and  sin  ^  {&'  -\~  &")  for  the  latter;  or, 

Ax^M—  log  ^f^.^  ~  ^/2  cos  I  (ir  +  ^^'0  =  M—J^       (sav) 

Jj/  =  M  J  log  ll^^j  sin  i  {^r  +  &")  =  M^A:        (say) 

For  the  entire  range  we  evidently  have 

X=  y  Jx  =r  M~  I  A^=M-^ 
K  g 

the  summation  extending  from  ^  =  ^  to  ^  =  w,  w  being  the 
angle  of  fall. 

To  determine  the  value  of  co  we  have,  since  the  sum  of 
the  positive  increments  of  ^'^  in  the  ascending  branch  is  equal 
(numerically)  to  the  sum  of  the  negative  increments  in  the 
descending:  branch, 

Expression  for  the  Time.— For  the  time  of  flight  we 
have,  when  dx  is  small, 

u 
in   which  u  is  the  mean  horizontal  velocity  corresponding 
to  Ax ;  but,  from  (15),  when  n  =  2, 
__  k 

whence 


\{i)-{^)\ 


I        Ax  ^  ( '^^  —  ^^^'^  ^  ^ 
At  = 


k 
or,  substituting  for  Ax  its  value  given  above, 


M  =  ^AZS,..      ,.A\ 


If  we  put 


je=Jf{(0-w[* 


12 


94  EXTERIOR    BALLISTICS. 

we  may  have 

log  J0=log  J^  +  i  log  [(0  -  i^)] 

The  two  values  of  log  [{i)  —  (^)]  corresponding  to  the 
extremities  of  the  arc  Js,  are 

log  [  (0 -(<?')].  and  log  [(0 -(#")] 
the  first  of  which   is  too  small  and  the   second  too  great; 
whence,  taking  their  arithmetical  mean, 

log  Je=\og  J?+ilog[{i)-{d')]  +  i\og[{i)-{9")l 
by  means  of  which  0  may  be  computed,  and  we  then  have 

Tables. — General  Otto,  of  the  Prussian  Artillery,  has 
published  extensive  tables*  of  the  values  of  {&),  q,  C,  and  6 ■— 
the  last  three  double  entry  tables  with  i  and  (p  for  the  argu- 
ments— by  means  of  which  it  is  easy  to  solve  many  of  the 
problems  of  high-angle  fire. 

Determination  of  k^, — General  Otto,  in  the  work 
above  cited,  gives  the  following  method  for  determining  k""  : 
We  have 


and 
whence 


an  equation  independent  o{  P. 

independent  of  X  and  T,  being  functions  of  the  angle  i  and 

the  angle  of  projection  cp ;  and  their  ratio  -^  may  be  tabu- 
lated with  these  angles  for  arguments.  General  Otto  has 
inserted  such  a  table  in  his  work  calculated  for  angles  of 

*  "  Tafeln  fiir  den  Bombenwurf."     Translated  into  French  by  Rieffel  with  the  title  "  Tables 
Balistiqucs  Generales  pourie  tir  eleve."     Paris,  1844. 


X^ 

g 

/ 

T'z=. 

MX 

e 

gT^" 

6' 

dent  oiU 

\     Moreover 

e 

and  6" 

are 

both 

EXTERIOR   BALLISTICS.  95 

projection  beginning  at  30°  and  proceeding  by  intervals  of 
5°  up  to  75°. 

Now,  suppose  a  certain  projectile  is  fired  with  a  known 
angle  of  projection  (p,  and  its  horizontal  range  X,  and  time 
of  flight    7",  are   carefully    measured.      With   this  data  we 

compute-^  by  means  of  the  above  equation;  and  entering 

Otto's  Table  III.  with  the   argument  ^5  find  in  the  proper 

column  the  computed  value  of -^,  and  take  out  the  corre- 

sponding  value  oil.  Next,  with  (p  and  ?  as  arguments,  take 
from  Table  IF.  the  value  of  ?^,  from  which  k^  can  be  computed 
by  the  following  formula,  derived  from  the  expression  for  X 
given  above  : 

^  ~  M  e 


bashforth's  method. 

For  all  values  of  n  greater  than  unity  the  differential 
equations  of  motion  take  their  simplest  form  when  ?/ =  3. 
For  this  reason  Professor  Bash  forth  assumes  the  cubic  law 
of  resistance  throughout  the  whole  extent  of  the  trajectory, 
and  employs  variable  coefficients  to  make  the  results  con- 
form to  the  actual  resistance. 

Making  ;/  ==  3,  equation  (25)  becomes 

-  k         d  tan  d- 

at  =^ 

g 


{  «  -  W  j  * 
in  which 

{&)  =  tan  &-\-^  tan^  ^ 

From  (14)  we  have,  when  ;2  =  3  and  ??  =  o, 

...       k' 

W  =  -3 

and  this  substituted  in  the  above  expression  for  dt  gives,  by 
a  slight  reduction, 


96  EXTERIOR    BALLISTICS. 

^^  _  _  3 d  tan  d^ 

^   {1-^^(3  tan  .^  +  tan^^)p 

Introducint^  Bashfortli's  coefficient  K,  making 


g  %v  Viooo/ 
to  correspond   with  his  notation,  and  integrating  between 
the  limits  (^,  d)  and  (o,  /),  we  have 

<^/       I  i_;k(3  tan/^  +  tan^^)i  *        ^      " 
Operating  in  the  same  way  upon  (26)  and  {2^)^  we  obtain 


1~7  ^^ 


z;,'     /^-^ d  tan  /> 

S  j       I  I  _  -^  (3  tan  />  +  tan^  if)  \ 

v''    r^  tan  d-  d  tan  d- 

y-irf  T^ — — T;T*  =  T^n 


and  ^..    ^  tan  d-  d  tan  ^  _  t^^ 

-y{2>  tan  ,>  +  tan^  (>)  U'        ^ 


Professor  Bashforth  has  published  extensive  tables  of  the 
definite  integrals  ''*7^,*X:J,and  '^F^^  for  values  of  ^  extending 
from  +60°  to  —  60°,  and  of  y  from  o  to  100,  calculated  by 
quadratures;  by  means  of  which  the  principal  elements  of  a 
trajectory  may  be  accurately  determined  as  follows: 

As  the  coefficient  of  resistance  K  generally  varies  with 
the  velocity,  the  trajectory  must  be  divided  into  arcs  of  such 
limited  extent  that  the  value  of  K  for  each  arc  may  be  con- 
sidered constant ;  and  it  should  be  so  taken  as  to  give,  as 
nearly  as  possible,  its  mean  value  for  the  arc  under  con- 
sideration. 

In  the  equation  given  on  page  65,  viz.: 

/loooX-'       /iooo\'   .    K  d"  {     ^  ,   ♦     .     I 

suppose  U  and  ^  to  be  the  initial  horizontal  velocity  and 
angle  of  projection  respectively,  and  both  known ;  and  let 
&,  also  known,  be  the  inclination  of  the  forward  extremity 


\a^. 


—  *•  »  TJ   J.    X 


EXTERIOR    BALLISTICS.  97 

of  the  first  arc  into  which  the  trajectory  is  divided.  Now, 
assuming  a  mean  velocity  for  this  arc,  take  out  the  corre- 
sponding value  of  K  from   the  proper  table  and   compute 

(I ooo\ ^ 
- — 1  ;  then,  in  the  same  equation,  changing  (p  to  />,  17  be- 

comes  tlie  horizontal  velocit}^  at  the  forward  extremity  ot 
the  arc,  which  can  also  be  computed. 

Next  compute  y  by  means  of  the  equation  given  above, 
with  which  and  the  known  values  of  ^  and  ^  enter  the 
tables  and  take  out  '^T^ ,  ^X^ ,  and  ^Y^-^   lastly,  multiplying 

2 

the  first  by  — ^,  and  each   of  the   others  by  -— ,  we  have  the 

^   g  ^  g 

time  of  describing  the  first  arc  of  the  trajectory  and  the  co- 
ordinates of  its  for  vvard  extremity.  By  repeating  the  process 
with  the  second  and  following  arcs  into  which  the  trajectory 
may  be  divided,  the  whole  trajectory  becomes  known. 

Professor  Bashforth  gives  various  other  tables  in  his 
work,  besides  those  we  have  mentioned,  for  facilitating  the 
calculation  of  trajectories  by  his  method,  with  examples  of 
their  application  and  full  directions  for  their  use. 

Modiflcatioii  of  Bashfortli's  Method  for  low  Velo- 
cities.— When  the  initial  velocity. does  not  exceed  790  f.  s. 
the  law  of  resistance  is  that  of  the  square  of  the  velocity  for 
the  entire  trajectory;  and  even  when  the  initial  velocity  is 
as  great  as  1000  f.  s.  examples  show  that  no  material  error 
results  if  we  still  retain  the  law  of  the  square  in  our  calcu- 
lations;  and  this  furnishes  a  very  easy  method  for  calcu- 
lating trajectories  for  high  angles  of  projection  and  for  the 
initial  velocities  usually  employed  in  high-angle  fire,  and 
which,  it  is  believed,  gives  as  accurate  results  as  by  any 
other  method,  and  with  less  labor. 

Making  ;/  =2,  equation  (25)  becomes 

k        d  tan  & 


dt 
in  which 


6   {(,-)_(^)|4 
((?)  =  !{  tan  d  sec  &  +  log  tan  g  +  |^)  | 


98  EXTERIOR    BALLISTICS. 

We  also  have  from  (15),  when  «  =  2,  and  ^  =  0, 

k^        I 

( 0  =  —  =  —    (say) 

and  this  substituted  in  the  above  expression  for  ^/ gives 
Vc.         d  tan  /> 


dt—  — 


?  ji_^(#)|i 


whence 

tan  ^  V, 


In  the  same  wa}^  we  obtain  from  (26)  and  (27)  the  follow- 


ing expressions  for  x  and  y  : 

d  tan  d-        v^ 


<    Pi 


r(^>) 


"^x,' 


^"^  -  .      ^</>  tan  ^  ^  tan  & 


It  will  be  seen  that  this  method  depends  upon  tables  of 
definite  integrals  which  must  be  calculated  by  quadratures 
as  in  Bashforth's  method,  and  with  the  same  number  of 
arguments;  but  the  great  advantage  of  these  formulas  over 
Bashforth's  is  in  the  fact  that  y  is  constant  for  a  given  tra- 
jectory, and,  therefore,  the  labor  of  calculation  is  the  same 
for  all  angles  of  projection. 

To  determine  the  value  of  k^  for  oblong  projectiles  of 
the  standard  type  we  have 

2A 

Taking  the  value  of  A  derived  from  the  Bashforth   experi- 
ments for  velocities  less  than  790  f.  s.,  and  making  ^^=1  32.16, 

-« fi"d  k'  ^  [5.4359033]  c 

For  the  Krupp  projectiles  we  should  have,  taking  May- 
evski's  value  of  A, 

'^'  =  [5-5367564]  c 

The  numbers   between  brackets  are  the  logarithms  of  the 
factors  by  which  C  is  to  be  multiplied. 


EXTERIOR   BALLISTICS.  99 

For  computing-  v^  we  have  from  (32),  when  ^  =  0, 

/(t/„)=-J(^)+/(f/)  (56) 

in   which  ^   may  be   the  inclination  at  any  point  in  either 

branch,  and  U  the  corresponding  horizontal  velocity.      The 

values  of  (^)  are  given  in  Table  III. 

To  show  the  practical  working  of  this  method,  we  will 

take    the    example    from     Bashforth    already    given    (see 

page  6j).     The  data  are:  V:^j^\   f.   s. ;  ^  =  30°;    ^  =  6.27 

inches,    and    w:=z'/o    lbs.;     whence    ^=650.385   f.    s.,    and 

70 
C^^-rp — -  ==  1.78059.     Determine  the  range,  time  of  flight, 

angle  of  fall,  and  terminal  velocity. 

First  compute  v^.     We  have  from  Table  III. 

(30°)  =  0.60799 
whence,  from  (56), 

^^^S~  "^  ""(^50.385)  =  0.68291  +  0.93354  =  I.61645 

therefore,  from  Table  I.,  ' 

^0  =  525.91  f.  s. 

Computation  of  y : 

log  (7=0.2505630 
constant  log  =  5 -435903 3 

log /^''  =  5.6864663 
log  z/q'  =  5.4418228 

log  r  =  97553565 

r  =  0.56932 

As  general  tables  of  the  definite  integrals  '^7'^,  '^X^ ,  and 
**  V^  have  not  yet  been  prepared,  the  following  table  has 
been  calculated  for  this  particular  example,  merely  to  illus- 
trate the  method  : 


lOO 


EXTERIOR   BALLISTICS. 


r  = 

=  0.56932 

0 

T 

X 

V 

30° 

0.63676 

0 . 70486 

0.21775 

24 

.47838 

.51493 

.12039 

18 

.34169 

.35965 

.06045 

12 

.21944 

.22662 

. 02460 

+  6 

+  .10673 

+  .10838 

+  .00575 

0 

.00000 

. 00000 

. 00000 

-  6 

-  .10358 

—  . 10208 

+  .00531 

12 

.20647 

.20061 

.02091 

18 

.31104 

.29793 

.04701 

24 

.41977 

. 39620 

.08479 

30 

•53551 

.49759 

.13656 

36 

.66179 

.60449 

.20615 

37 

.68417 

.62303 

.21987 

The  value  of  ^°°V°  by  the  above  table  is  0.21775,  and  as 
this  must  be  equal  to  °F"  we  see  at  a  glance  that  co  lies 
between  —  36°  and  —  37° ;  and  by  interpolation  we  get 
w=r— 36°5i';  and  therefore  °X'J  =  0.62025  and  ^T'J  0.68081. 
Adding  to  these  the  numbers  corresponding  to  the  argument 
30°,    we    get   "PX-  =  1.32511,    and   *^7:y^  =  I.3I757-      Lastly, 

multiplying  the  first  of  these  by  -^,and  the  second  by  — ,  we 

obtain 

X=  11396  ft. 
and 

r=  21^546 

which  agree  with  Bashforth's  calculations. 

The  terminal  velocity  is  found  from  (32),  viz.: 


and 


V...  =  «,..  sec  CO 


We  find 


and 


^o.  =  434.7  f.  s. 
^«o  =  543-2  f.  s. 
It  will  be  seen  that  the  inverse  problem,  namely,  Given 


EXTERIOR  BALLISTICS.  lOI 

the  terminal  velocity  and  angle  of  fall,  to  determine  the 
initial  velocity,  angle  of  projection,  range,  and  time,  can  be 
solved  by  this  method  with  the  same  ease  and  accuracy  as 
the  direct  problem.  We  should  first  compute  the  summit 
velocity  by  the  equation 

/W  =  /(0-|=H  (57) 

and  then  all  the  other  elements  would  be  determined,  as 
already  explained. 

In  calculating  trajectories  by  this  method  with  the  help 
of  tables  of  the  definite  integrals  '^T^ ,  etc.,  it  will  generally 
be  necessary,  as  in  Bashforth's  method,  to  interpolate  with 
reference  to  y  as  well  as  d-,  and  for  this  purpose  the  integrals 
must  be  tabulated  for  different  values  of  y  proceeding  by 
constant  diff'erences,  and  including  the  highest  and  lowest 
values  of  y  likely  to  be  needed  in  practice,  which  are,  ap- 
proximately, I  and  O.2. 


13 


CHAPTER  VII. 

TRAJECTORIES   CONTINUED — DIRECT   FIRE. 

Niven's  Method. — If  a  is  some  mean  value  of  sec  d 
between  the  limits  of  integration  ;  that  is,  if  we  make 

a  =  sec  ^?         (say) 

then  equations  (17)  to  (20)  may  be  written  as  follows: 

_   6;  d  {a  li) 

^  A      (a  uf 

C  -T-d  (a  m) 

^^=-^cos^-^^^^^  (58) 

C     .     -.   d{au) 

dy—  —  -7-  smd  .      .„_\ 

A  {a  lif  ' 

C    d  (a  u) 

A     (auf-' 

Making  a  u=iu\  and  integrating  so  that  t,  x,  y,  and  s 

shall  each  be  zero  when  u'  =  U\  we  have 


t  =  —£_\_l L_l 


C 


cos  i)^ 


y  = 


{n-2)A  "-''^^    (  u'^-' —  U'""-- 

c 


11  — 2)  A  \   u"-'        U'"^  ) 


0 

Comparing  these  equations  with  those  deduced  in  Chap- 
ter IV.  for  rectilinear  motion,  it  will  be  evident  that  we 
have  as  follows: 

t  =  ciT{u^)-  r{U')-]  (59) 

x=Cco?>^[S  {u')  -  S  ( U')~j  (60) 

y=C  sin  J  [S  (?/)  -  S{U'\^x  tan  5  (61) 

s=ClS{u')-S{U')']  (62) 


EXTERIOR   BALLISTICS.  IO3 

The  first  three  of  these  equations  (or  their  equivalents) 
were  first  published  by  Mr.  Niven  in  1877,  and  in  connection 
with  equation  (38),  viz.: 

D=Ccos.J[D  {u')  -  D  ( U')]  (63) 

constitute  what  is  known  as  "Niven's  Method." 

If  we  use  the  /-function  instead  of  the  Z>-function,  equa- 
tion (63)  becomes 

/}  =  25_^  cos  &U  W)  -  I  ( U')-\  (64) 

or,  better  still,  for  direct  fire  (see  Chapter  V.), 

D  =  25_^  sec  (f  [/  (u  sec  f)  -  I  { F)]  (65) 


in  which 


log  ^=:l. 45  70926* 


The  values  of  ^  adopted  by  Mr.  Niven  are  as  follows: 
For  the  /^-integral 

-—       tan  <p  4-  tan  ^ 
tan  *.  =  —^ 

For  the  X-,  V-,  and  T-integrals 
-       -        U 

for  the  ascending  branch,  and 

U  +  ic        3 
for   the    descending    branch    of    the    trajectory.      For   the 
method  of  deducing  these  expressions  for  ^,  see  a  paper  by 
Professor  J.  M.  Rice,  U.  S.  Navy,  in  the  eighth  volume  of 
"  Proceedings  Naval  Institute,"  page  191. 

We  will  now  apply  these  formulae  to  the  solution  of  a 
problem  of  direct  fire;  and,  as  we  wish  to  compare  the  re- 
sults obtained  with  those  to  be  deduced  from  other  methods 
we  will  use  Table  I.  of  this  work  instead  of  Niven's  tables, 
and  we  will  also  perform  the  calculations  with  more  accu- 
racy than  is  generally  necessary  in  practice. 


104  EXTERIOR   BALLISTICS. 

Example  of  Niven  s  Method. — A  1 2-inch  service  projectile 
is  fired  at  an  angle  of  departure  of  io°,  and  an  initial  velocity 
of  1886  f.  s.  Find  v,  x,  y,  and  /  (a)  when  &  =  o,  and  (b) 
when  &  =  —  1;^°. 

Here  d=  12  in.,  z£'=  800  lbs.,  C  — ,<c^  10°,  V^  1886 

144 

f.  s.,  U—  1886  cos  10°  =  1857.33. 

(a)  ??  =  o     .-,  n=  10°.     We  have  first 

tan  ?^,  =  i  tan  10°  =  0.0831635 
.-.  5,  =  5°  2'  i8'^  and  U' =  Usgc'J,=^  1864.56 
Next  compute  u'  by  means  of  the  equation 

nu')  =  ^^sec&,  +  I{U') 

or 

/  (u^)  =  0.06308  +  0.03624  =  0.09932 

.-.  u'  =  1328.96  =  ?^  sec  d-^ 
.-.  ti^=  132372 

Next  compute  the  value  of  ^  to  be  used  with  the  X-, 
F-,  and  ^-integrals.     We  have 

^  =  5°  2'  18''  +  ^^57-33  -  132372  X  1?  ..  5°  35'  51'' 
^1857.33+132372        3 

The  new  values  of  U^  and  1/  are,  therefore, 

U'  =  1866.25,  and  u'  =  1330.06 

From  Table  I.  we  find 

5(^/0-2855.3  5  (?0  =  5239-2 

'    T{U')=  1.258  T{u')=    2.778 

.•.4  =  ^{2.778-  1.258}  ::=8^444 

■^o  =  Y^cos5|  5239-2  -  2855.3  |  — 13180.7  ft. 

yo  =  ^  tan  d^  =  1291.8  ft. 

(b)  ^=  —  13°.     It  will  be  necessary  in  this  case  to  take 
a  new  origin  at  the  summit  of  the  trajectory,  as  thei'e  is  no 


EXTERIOR   BALLISTICS.  I05 

provision  made  in  this  method  for  calculating  an  arc  of  a 
trajectory  lying-  partly  in  the  ascending  and  partly  in  the 
descending  branches.  Indeed,  since  the  differential  ex- 
pression for  J/  contains  sin  ^  as  a  factor,  which  becomes  zero 
at  the  summit  and  changes  its  sign  in  the  descending  branch, 
equation  (61)  does  not  hold  true,  unless  the  limits  of  integra- 
tion {if  and  d^)  are  both  positive  or  both  negative. 

We  have,  then,  for  this  arc  of  the  trajectory  the  follow- 
ing data  : 

F=  U^  1323.72,  ip  =0°,  ^=  -  13°,  and  D=  13° 

tan  ^j  =  —  i  tan  13°=  —  o.ii5434i_  .'  .^,=z  —  6°  35'  s" 

[/'=  1332. SI      7(^0  =  0.09860 

7  {u')  =  0.08222  +  0.09860  =  0.18082 

.' .  u'  —  1064.39  =  Vq  cos  d^  sec  d^^ 

.  • .  vq^  1085.18,  and  Uq  =  1057.37 

^  :^  -  6°  35'  ^"  -  ^32372-1052:37      jz_I3  _  _60  6'o'' 
1323.72  +  1057.37  3 

The  new  values  of  U^  and  u^  are,  therefore, 

f/^=  1331.26,  and  u'  z=  1063.39. 

From  Table  I.  we  get 

5  {[/')  =  5232.9  5  («0  =  701 1.7 

tIu')  =  2.773  ^M  =  4-282 


.•.^  =  ^{4.282-2.7731  =  8^383 

X  =  —  cos  ??  j  7011.7  —  5232.9  [  =  9826.3  ft. 


144 

y  =z  X  tan  ??  1=  —  1050. 1 
The  co-ordinates  of  the  point  of  the  trajectory  whose  in- 
clination is—  13°,  taking  the  origin  at  the  point  of  projec- 
tion, are  therefore 

X=  1 3 180.7  +  9826.3  =  23007.0  ft. 
F==    1291.8  —  1050.1  =      241.7  ft. 
And  the  time, 

7^=8.444  +  8.383  =  16^827 
For  comparison  we  have  computed  the  same  elements 


I06  EXTERIOR   BALLISTICS. 

directly  from  equations  (i6),  (25),  (26),  and  (27),  dividing  the 
whole  arc  into  three  parts,  with  the  points  of  division  corre- 
sponding to  velocities  of  1330  f.  s.  and  1120  f.  s.  respectively. 
The  integrals  for  each  arc  were  computed  by  quadratures, 
and  the  following  are  the  final  results: 

^^=1081.55    f.   s.;   Xi=  23025.7  ft.;    F=:  243.14    ft.,   and 

r=  16^843. 

The  agreement  between  these  two  sets  of  values  is  re- 
markably close,  and  shows  that  for  the  purpose  of  com- 
puting co-ordinates  of  different  points  of  a  trajectory, 
Niven's  method  is  all  that  could  be  desired  so  far  as  ac- 
curacy is  concerned.  For  high  angles  of  projection  the 
trajectory  should  be  divided  into  arcs  not  exceeding  10°  or 
15°  each,  and  always  with  one  point  of  division  at  the  sum- 
mit. 

Example  2. — Given  d  ^  12  in.,  'w=^  800  lbs.,  F=  1886  f.  s., 
and  ip  =  30°.  Compute  the  time  and  co-ordinates  when 
d-  =  24°. 

Answer : 

BY    niven's    method.  BY    QUADRATURES. 

^,  =   27°  4'   29'' 

"5  =  2f   19'  4" 

Xq  =  8482.0  ft.  8481.4  ft. 

_;/,  =  4381.2  ft.  4381.9  ft. 

/^  =  5''.889  5^888 

Vff  =  1400.58  f.  s.  1400.4  f.  s. 

In  the  same  manner,  by  successive  steps,  can  the  whole 
trajectory  be  computed.  In  practice  it  is  never  necessary 
to  divide  a  trajectory  into  arcs  of  less  than  10°. 

Sladen's  Method  for  Low-Aiigle  Firing/'^  — When 
the  angle  of  projection  is  small,  say  not  exceeding  3°,  the 
time  corresponding  to  a  given  range  can  be  computed  with 
great  accuracy  by  means  of  (29)  and  (30).  We  should  first 
find  V  by  means  of  the  equation 

*  "Principles  of  Gunnery,"  by  Major  J,  Sladen,  R.A.,  London,  1879,  Chapter  VI. 


EXTERIOR    BALLISTICS. 


107 


and  then  with  th.is  value  of  v  compute  T  by  means  of  (29). 
In  the  same  manner  we  could  find  the  value  of  /  for  a  given 
value  of  Xy  less  than  X ;  and  these  values  of  T  and  t  substi- 
tuted in  (46),  viz., 

would  give  the  value  of  jj^  corresponding  to  x ;  since,  under 
the  conditions  supposed,  the  vertical  component  of  the  velo- 
city would  be  so  small  as  to  produce  no  appreciable  resist- 
ance to  the  projectile  in  that  direction. 

Example  i — Required  the  following  co  ordinates  of  the 
trajectory  described  by  a  500-grain  bullet  fired  from  a 
Springfield  rifle,  for  a  range  of  600  ft.,  viz. :  when  ;ir=  150  ft., 
300  ft.,  and  450  ft.  respectively  ;  r^  =  524.29,  ^^  =  534.22. 

Here  ^==  0.45  in.,  zv  =  500  grains  =  J^  lb.,  F=  1280  f.  s., 
and  X=  600  ft.  We  first  find  5  (K)  =  5509.70;  r(F)  =  2.985; 
and 

i\  X  534.22 


C 


0.35942 


(0.45)  X  524-29 

The  principal  steps  of  the    remaining    calculations  are 
given  in  the  following  table : 


(ft.) 

X 
C 

S{v) 

(f.  s.) 

/ 

(inches.) 

(inches.) 

(inches.) 

150 

417-34 

5727.04 

1209.72 

0".  12055 

9-365 

9.406 

7-950 

300 

834.69 

6344-39 

1146.76 

0".  24814 

13-167 

12.987 

10.600 

450 

1252.03 

6761.73 

I09I .31 

0". 38235 

10.386 

9-956 

7-950 

Coo 

1669.38 

7179-08 

1046.55 

0".  52313 

{T) 

0.000 

0.000 

0.000 

The  sixth  column  gives  the  computed  values  oiy,  and  the 
seventh  the  mean  of  five  trajectories  measured  with  great 
care  at  Creedmoor  by  Mr.  H.  G.  Sinclair,  in  charge  of  the 
"  Forest  and  Stream  Trajectory  Test."  The  last  column 
gives  the  corresponding  values  of  j  in  vacuo,  computed  by 
(45)- 


t08  EXTERIOR   BALLISTICS. 

SIACCl'S   METHOD   FOR   DIRECT   FIRE. 

dy 
Expression  for  y. — We  have  from  (35),  since  tan  ^  =  -7^ 

dy ^^  a  C 

dx 
or 

2     j   dy 


tan^-^{/(«')-/(^')[ 


^{-^-ta„,}-/(i/')=-/(«') 
We  also  have  from  (58) 


a    .  du' 

ax  =: 


C  A  u'^-' 

whence  multiplying  the  last  two  equations  together,  mem- 
ber by  member, 
2 

C 

Integrating  and  making  x  and  j  both  zero  at  the  origin, 
where  //'  =  U\  we  have 


{</^-u„,^.(-|./(^V-  =  ^|3^ 


2   j  ^  \         a   ^,^„^  I    fl{u')du' 


Making  for  convenience 


-.K)=i/^^' 


(in  which  the  ^'s  must  not  be  confounded)  the  above  equa- 
tion becomes 

^,{>/-^tan^}   -^I{U')x=-  \a{u')-A{U')\ 

From  (60)  we  have 

^x  =  S{u')-S{U') 

whence,  by  division, 

^   \y     nn<.l      nu'\-     A{u')-A{U') 
_|__tan^}-/(f/)_-  ^-(„,)_5(f;,) 

aC  iA(u')-A(C/')        ,,rml  iA^\ 


or 

z 

X 


EXTERIOR  BALLISTICS.  IO9 

Calculation  of  the  ^-Function.— We  have  (Chap- 
ter V.) 

and  therefore 

_  g  Q  , 

^  l2\n-  I)  A'  u"^'^  +  {n-2)Au'^-'  +  ^' 
which  becomes,  when  ;/  =  2, 

The  constants  Q,  corresponding  to  the  five  different  ex- 
pressions for  the  resistance,  are  given  in  Chapter  V.,  and 
the  values  of  Q'  are  to  be  determined  as  explained  in  Chapter 
IV.  Making  the  necessary  substitutions,  and  using  A  {v)  as 
the  general  functional  symbol,  we  have  for  standard  oblong 
projectiles  the  following  expressions  for  calculating  the  A- 
functions : 

2800  f.  s.  >  -t^  >  1330  f.  s.  : 

A  {v)  =  [8  9012292]  -^,  +  [2.6701589]  log  v  -  1714-55 

1330  f.  s.  >  z'  >  II 20  f.  s. : 

A  (7;)  =  [14.6562945]  ^  +  [5.1480576]  i  -  53.13 

1 1 20  f.  s.  >  z'  >  990  f .  s. : 
A  (v)  =  [32.2571789]  ^„  +  [14.4412953]  ^  +  126.68 

990  f.  s.  >  z^  >  790  f .  s. : 
A  (v)  =  [14.9781903]  — ,  —  [5.9124902]  ^  +  449.89 

790  f.  s.  >  -z/  >  100  f.  s. : 
A  {v)  =  [9.6655206]  ^  +  [4.1438598]  log  V  -  45916.40 

The  values  of  A  {v)  calculated  by  the  above  formulae  are 
given  in  Table  I. 
14 


no  EXTERIOR   BALLISTICS. 

Equation  {66),  together  with  (35),  (59),  and  (60),  are  the 
fundamental  equations  of  *'  Siacci's  method."  This  method, 
by  Major  F.  Siacci,  of  the  Italian  Artillery,  was  published 
in  the  Revue  d' Artillerie  for  October,  1880.  A  translation 
of  this  paper  by  Lieutenant  O.  B.  Mitcham,  Ordnance  De- 
partment, U.  S.  A.,  was  printed  in  the  report  of  the  Chief 
of  Ordnance  for  1881.  Lieutenant  Mitcham  added  to  his 
translation  a  ballistic  table  adapted  to  English  units,  and 
based  upon  the  coefficients  of  resistance  deduced  by  Gene- 
ral Mayevski  from  the  Russian  and  English  experiments 
noticed  in  Chapter  IL  In  this  table  he  gives  for  the  first 
time  the  values  of  T{v). 

We  will,  for  convenience,  collect  thesd  equations  to- 
gether and  renumber  them : 

They  are : 

tan  ^  -  tan  ??  =  ^  I  /  (;/)  —  I{U')  |  {^7) 

x^^-\s{u')-S{U')\  (68) 

: — :  I  >i  1 1    in  —    ^ 

S{u')  —  i,{U') 

t  =ClT{u')-T{U')^  (70) 

u'  —  av  cos  d^  (71) 

As  the  origin  of  co-ordinates  is  at  the  point  of  departure, 
y  is  zero  at  the  origin  and  also  at  the  point  in  the  descend- 
ing branch  where  the  trajectory  pierces  the  horizontal  plane 
passing  through  the  muzzle  of  the  gun.  Calling  the  velo- 
city at  this  point  v^,  we  shall  have,  making  —  d^  z:z  w, 

u'o>  =  «  ^  ^  cos  CO  (72) 

From  (69)  we  have 

aC  \A  (u'^)-A  (U')         ..  ^„^  )  ._s 

^  2      [   S  {u'^)  —  S  {[/')  ^  ) 

and  from  {6y) 

a  C 


~        ^        2   \s{u')  —  :^{U')     ^'^^M      ^  ^^ 


tan 


^  =^  "^  I  /  {ti'^)  -  7(^0  I    -  tan  CO  (74) 


EXTERIOR    BALLISTICS.  Ill 

Eliminating^  tan  ^  from  these  last  two  equations  gives 

tan  ^^  =  —  I  ^  (^^  ^)  -  ■5l/?.)-5(^0  ^ 
From  (68)  and  (70)  we  have 

X=z—  \s{2/^)-S{U')\  (76) 

and  a    I      ^      ^  ' ) 

T=  C[T(u'^)^T{U')]  {77) 

By  means  of  equations  {67)  to  {77)  all  problems  of  ex- 
terior ballistics  in  the  plane  of  fire  may  be  solved.  If  we 
wish  to  compute  the  co-ordinates  of  the  extremities  of  any 
arc  of  a  trajectory  having  the  inclinations  (f  and  d^,  we  should 
make  use  of  equations  {67)  to  (71).  If  the  object  is  to  deter- 
mine the  elements  of  a  complete  trajectory  lying  above  the 
horizontal  plane  passing  through  the  muzzle  of  the  gun,  at 
one  operation,  we  should  employ  equations  (72)  to  {77).  We 
will  give  an  example  of  each,  using  Didion's  value  of  a. 

Example  i. — Given    F=  1886  f.  s. ;   <^=  12  in.;  2£;  =  800 
lbs.,  ^  =  10°,  and  ??  =  — 13°;  to  find  z/^,  ;ir0,  je,  and /e.     (See 
example  i,  Niven's  method.) 
We  have  first 

(10°) +  (13°) 

a  =  — ^ i  \   I  o  =  1.00723 1 

Next  ^^"  10°  +  tan  13°  ^  ^ 

U'  =  1886  «  cos  10°  =  1870.78 

From  Table  I., 

5(^0=-2838.3;zJ  (^0=44.06;  /(^0=O-O358i;  r<^U')^i.2t,o 

From  {67^  we  have 

/(^/.)  =^  1  ^"'^^  '""^  +  ^""^  '^°  1   +^(^') 

=  0.14554  +  0.03581  =0.18135 

.-.  2^'=:  1063.42;  5(/0==7Oii<4; -^K)=440.44;  7 V)  =4-282. 

These  values  substituted  in  (68),  (69),  and  (70)  give 

xq^=.  23017  ft. 

yQ  =  248.06  ft. 

tQ  z=  i6''.844 


112  EXTERIOR    BALLISTICS. 

From  (71)  we  have 

Ve  = K  =  1083.6  f.  s. 

a  cos  a 

These  results  are  quite  as  accurate  as  those  deduced  by 
Niven's  method  by  two  steps. 

Example   2. — Required     the    horizontal    range,    time    ot 
flight,  and  striking  velocity,  with  the  data  ot  Example  i. 

In  computing  «  we  will  assume  an  angle  of  fall  of —  14°  30', 
which  gives 

«==  1.008645 

.-.  ^'=1873.40 
5(^0=2828.5;  A{U')=Al-7^\  7(^/0=0.03563;   n^0=i-243- 
From  (73)  we  have 

^^^^g|i=^^tan,  +  /(t/')  =  o.09856 

from  which  to  calculate  ic'^.  As  the  relation  between  the 
S-function  and  y^-function  does  not  admit  of  a  direct  solu- 
tion of  this  equation,  it  will  be  necessary  to  determine  the 
value  of  z/o,  by  successive  approximations;  and  for  this  pur- 
pose the  rule  of  **  Double  Position"  is  well  adapted.  This 
rule  is  deduced  as  follows  :  Let  u^  and  u^  be  two  near  values 
of?/  (or  the  quantity  to  be  determined),  one  greater  and  the 
other  less  ;  and  e^  and  e^  the  errors  respectively,  when  n^  and 
u^  are  substituted  for  u  in  the  equation  to  be  solved.  Tiien, 
upon  the  hypothesis  that  the  errors  in  the  results  are  pro- 
portional to  the  errors  in  the  assumed  data,  we  have 

e^\  e^W  u  —  //j  :  u  —  u^ 

whence,  by  division, 

e^—  e^\  e^W  u^  —  u^'.  u  —  u^ 
or 

e^  —  e^\  e^\ :  u^  —  u^\  ii  —  ti^ 

from  which  is  derived  the  following  rule:  As  the  difference 
of  the  errors  is  to  the  difference  of  the  assumed  numbers,  so 
is  the  lesser  of  the  two  errors  (numerically)  to  the  correc- 
tion to  be  applied  to  the  corresponding  assumed  number. 


EXTERIOR    BALLISTICS.  II 3 

If  11^  and  u^  are  selected  with  judgment,  the  resulting 
value  of  II  will  generally  be  sufficiently  correct  by  a  single 
application  of  the  rule,  or,  at  most,  by  two  trials. 

In  our  example  assume  it^  =z  1050,  for  a  first  trial ;  whence 
5  (1050)  =  7143.7,  and  A  (1050)  =464.94 ;  and  these  in  the 
above  equation  give 

464.94  —  43.71  ^ 

^  ^  ^        -to  /      _  0.09762 
7143.7-2828.5 

If  we  had  taken  for  71^  the  correct  value  of  u^^,  the  second 
member  would  have  been  0.09856,  and  hence  ^,  =  —  0.00094. 
Whenever  ^,  is  negative  the  assumed  value  of  u' ^  is  too 
great;  we  will,  therefore,  next  suppose  2/2=  1040,  and  pro- 
ceeding in  the  same  way  we  find  ^^  = +0-00128.  The  cor- 
rect value  of  u^^  is,  then,  between  1050  ft.  and  1040  ft.  Ap- 
plying the  rule,  we  have  the  following  proportion  : 

222  :  10  ::  94  :  4.23 

consequently  u'^  =  1050  —  4.23=  1045.77  f-  s. :  and  this  satis- 
fies the  above  equation. 
We  next  find 

5(?/^)  =  7i87.i;  ^  (//^)=473.2o;  7(2^'^) =0.1 9 154;  T{u'^)=4.44S 
We  now  have  from  (75) 

tan  io  = j  o.  191 54  —  0.09856  i  =  0.2605 1 

.-..(>=  14"  36^     (By  Table  III.) 
From  {26)  and  (yy) 

I  7187.1  —  2828.5  I  =24007  ft. 

r=  c:[  4.448  -  1.243]  =  i7".8o6 
From  (72) 

^«  = ^=i07i.4f.  s. 

a  cos 

Various  other  problems  may  be  solved  by  a  suitable  com- 
bination  of  equations  {6y)  to  (71).      Indeed,  if  a  velocity, 


a 


114  EXTERIOR   BALLISTICS. 

either  initial  or  terminal,  and  one  other  element  be  given, 
all  the  other  elements  may  be  computed,  though  in  certain 
cases  this  can  only  be  accomplished  by  successive  approxi- 
mations. Most  of  these  problems,  for  direct  fire,  will  be 
solved  further  on. 

Api>licatioii  of  Siacci's  Equations  to  Mortar- 
Firing. — For  low  velocities,  such  as  are  used  in  mortar- 
firing,  we  may  take  for  a  in  all  cases  the  following  value  : 

tan  ^ 

This  simplifies  the  calculations,  and  gives  results  sufficiently 
accurate  for  most  practical  purposes,  as  the  fbllowing  ex- 
amples will  show  : 

Example  i.— Given  F=75i  f.  s. ;  ^  =  30°;  and  log  C zizz 
0.25056.  Required  X,  T,  w,  and  v^.  (See  Example  i,  Chap- 
ter VI.) 

We  have,  Table  III.,  {(f)  =  0.60799. 

log  {(f)  =  9.78390 
log  tan  <p  =  9.76144 

log  a  ==  0.02246 

log  V=:  2.87564 

log  cos  ^  =  9.93753' 

log  U'  =  2.83563  U'  =  684.90 

5(£/0=i368i.i;  ^(^0-=  344443;/ (^0=0-80679;  T{U')=^ 
12.274. 

log  2  =  0.30103  [Equation  (73)] 

c.  log  a  —  9.97754 
'  c.  log  6'=  9.74944  (^dd  log  t^"  f) 

log  0.61 581  =9.78945 
7  (£/')  =  0.80679 

1.42260 

•    ^  (»'.)- 3444-43  ^,fo 
5(«'„)— 13681.1         * 


EXTERIOR   BALLISTICS.  II5 

By  double  position  we  find  from  this  equation 
7/^  =  45978 
.-.  5  (?/<,)  =  20443.1  ;  /(?/'^)  =  2.22481  :    T {2/^)  =24,4.04 

X=— I  20443.1  -  13681.1  I   =11434  ft. 

Tzzz  (7  [24.404  —    12.274]  =  2  I ''.60 

tan  (o  = j  2.22481  —  1.42260  [  [Eq.  (75)] 

.•...  =  36°  57' 

?/ 

^co  = ~ — =  546.3  f.  s.  [Eq.  (72)] 

a  cos  CO       -^^    ^  L    n   \/    /J 

Example  2. — Given    f^==  977.71    f.    s.,    ip  —  35°  21',    and 
log  C'^:  0.38722.     Required  X,  7",  w,  and  t/^^.     (See  Example 
3,  Chapter  VL) 
Answer: 

^=19328  ft. 
^-31^63 
?/'„  =517.63 
CO  =  44°  44' 
v^  =1675.65  f.  s.  ^  ^ 

Example  3. — Given  F:^  609.63  f.  s.  ;  ^  =  45°,  and  log  6^  = 
0.56809;    required  X,    T,  oj,  and  ?^^.     (See  Example  5,  Chap- 
ter VI.) 
Answer: 

X=  11984  ft. 
r=  28^30 
?/a.  =  436.52 
CO  =  49°  10' 
7;^  =  581.64 

Siacci's  Equations  for  Direct  Fire.— As  already 
stated,  a  is  some  mean  value  of  the  secants  of  the  inclina- 
tions of  the  extremities  of  the  arc  of  the  trajectory  over 
which  we  integrate ;  and  consequently  if  we  take  the  whole 

r 


Il6  EXTERIOR   BALLISTICS. 

trajectory  lying  above  the  level  of  the  gun,  a  will  be  greater 
than  I  and  less  than  sec  co.  To  illustrate,  suppose  we  have  for 
our  data  a  given  projectile  fired  with  a  certain  known  initial 
velocity  and  angle  of  projection,  and  we  wish  to  calculate 
the  angle  of  fall,  terminal  velocity,  range,  and  time  of  flight. 
If  we  calculate  these  elements  by  means  of  (75),  {ji),  (76), 
and  {j']^^  making  a  =  i,  they  will  be  too  great ;  while  if  a  is 
made  equal  to  sec  co,  or  even  sec  ^,  they  will  be  too  small ; 
and  the  correct  value  of  each  element  would  be  found  by 
giving  to  a  some  value  intermediate  to  the  two.  Moreover, 
the  value  of  a  which  would  give  the  exact  range  would  not 
give  the  exact  time  of  flight  or  terminal  velocity.  These 
principles  are  further  illustrated  by  the  follow^ing  numerical 
results,  calculated  from  the  data,  F=  1404  f.  s. ;  ^=10°; 
w  z=z  183  lbs.,  and  <^=  8  in. : 

a  =  1  a  =  sec  <p 

X=  13752  ft.  X—  13622  ft. 

v^  =  892.2  f.  s.  v^  =881.4  f.  s. 

co=-ifif  fti  =  -i3°23' 

T=if.04  T=12\SS 

As  the  true  values  of  these  elements  lie  between  those 
we  have  computed,  it  will  be  seen  that  either  set  of  values 
is  correct  enough  for  most  purposes.  It  is,  therefore,  ap- 
parent that  in  direct  fire  we  may  give  to  a  that  value  which 
shall  reduce  the  above  equations  to  their  simplest  forms,  pro- 
vided it  lies  between  the  limits  a=  i  and  a  =  sec  (p. 

As  we  have  already  seen  (Chapter  V.\  Major  Siacci 
gives  to  a  the  value 

n-2 

a  =  (sec  (f)  «^i 
by  means  of  which  equation  (37)  was  obtained,  viz.: 

tan^  =  tany-— J- j /(«')- /(F)}  (78) 

in  which 

,  cos  ^ 

cos  (p 


EXTERIOR   BALLISTICS.  II7 

Making  the  same  substitution  in  (68),  (69),  and  (70),  they 
become  respectively 

x-  =  C[S{u')-S{V)-]  (79) 

;i'  ^2  cos'  (f  {   S  (?/)  —  5  ( F)  ^     M       '     ^ 

When  ^  and  d-  are  so  small  that  the  ratio  of  their  cosines 
does  not  differ  much  from  unity,  we  may  put 

and  the  above  equations  become 

tan  &  =  tan  <p ^  ^  ^  (^)  -  /(H  [  (82) 

^      2  cos  <p  {        ^         ^    M 

;r=6^[5(7;)-.V(F)]  (83) 

-^'-tanc  6-      •i^e.)-^(F)  [  ,3. 

--tan  <p  -  ^  ^^^^,  ^  j  ^^^^-^_-  -  /(^^)  j  l«4j 

/-=— ^]  r(^)-r(F)i  (85) 

cos  ^  (      ^  ^         ^    M 

We  shall  retain  this  form  of  the  ballistic  equations  in 
what  follows,  though  when  very  accurate  results  are  de- 
sired we  must  use  ?/  instead  of  z^ 

When  J/ =  0,  we  have  from  (84) 

Substituting  for  tan  (p  in  (84)  its  value  from  (82),  and  re- 
ducing, we  have,  when  j'  =  o, 

2  cos=  <p  tan  «  =  C  I  /  (V)  -  ^  ^^.^  _  ^  ^y^  \ 
For  small  angles  of  projection  we  may  put 

2  cos'''  (f 
and,  therefore. 


cos  <p 

2  cos   (p  tan  o)  =  2  sui  co  cos  co ~  =  sui  2ft> 

^  cos  ^6» 


1  A{v)-A(V)l  ,     . 

sin  2<.  =  C-  I  / (.•)  -  s{v)-S{V)  \  ^^7) 

For  the  larger  angles  of  projection  employed  in  direct 

^5 


Il8  EXTERIOR   BALLISTICS. 

fire,  if  accurate  results  are  desired,  we  must  determine  (o  by 
the  equation 

tan  CO  =  tan  w ^  \  I {v)  -  I  (V)  \ 

^  2  COS    if    \      ^   ■'  ^      ^  \ 

using  ?/  instead  oi  v,  as  already  explained. 

Practical  Applications. — We  will  now  apply  Siacci's 
equations  to  the  solution  of  some  of  the  most  important 
problems  of  direct  fire. 

Problem  i. — Given  the  initial  velocity  and  angle  of  pro- 
jection, to  determine  the  range,  time  of  flight,  angle  of  fall,  and 
terminal  velocity. 

We  have  [equation  (86)] 

A{:v)-A{y)  _  sin  2^ 
S{v)--S{l^)  C      "T"     ^^^ 

from  which  to  calculate  v  by  ''  Double  Position,"  as  already 
explained.  Having  found  v^  the  remaining  elements  are 
computed  by  the  equations 

x=c[se.o-5(F)] 

r=— ^  I  T{v)-T{V)\ 
COS  ^  \  '  '  ) 

For  curved  fire  we  may  proceed  as  follows:  We  have, 
from  the  origin  to  the  summit, 

Now,  if  we  assume  tiiat  the  time  from  the  point  of  pro- 
jection to  the  summit  is  one-half  the  time  of  flight,  we  shall 
have,  from  the  above  expressions  for  7' and  4, 

r(7;)  =  2  T{v^-T{y) 
which  gives  z^  by  means  of  the  7^-functions,  v^  being  computed 
bv  the  equation 

derived  from  (82). 

Example  i. — The  8-inch  rifle  (converted)  fires  an  ogival- 


EXTERIOR    BALLISTICS.  II9 

headed  shot  weig-hing  183  lbs.  If  the  angle  of  projection 
is  10°,  and  the  initial  velocity  1404  f.  s.,  find  the  range,  time 
of  flight,  angle  of  fall,  and  terminal  velocity. 

We    have    F^  1404  f.   s.  ;  ^=zio°;  ze;  =  183   lbs.;  d=Z 
inches,  whence  log  C  —  0.45627 :  to  find  X,  T,  od,  and  v. 
From  Table  I.  we  find 

5(F)  =  4878.6 -0.8  X  25.1  =4858.5 
A  {V)  =  163.96  —  0.8  X  2.16  =  162.23 
/  (F)  =  0.08661  —  0.8  X  0.00082  =  0.08599 
T{V)  =  2.514—0.8  X  0.018=2.500. 
Next  compute  v: 

log  sin  2^  =  9.53405 
log  (7=0.45627 

log  o. II 96 1  =  9.07778 
/(F)  =  0.08599 

0.20560 

The   value  of  v  satisfying  this  equation  is  found  to  be 
V  =  873.8  ft.,  whence 

5  {v)  =  9641.8  A  (z/^)  =  1 145.65 

/  (v)  =  0.36668         T  {v'^)  =  7.030 
X,  T,  (0,  and  z'  are  now  computed  as  follows  : 

log  C  =  0.45627 
log[S(2.)- 5(F)]  =  3^7973 

log  X=  4.13600' 

X=  13677  ft.  ==4559  yc^s. 

\og[T{vy-  r(F)]  =  0.65610 

log  sec  <p  =  0.00665 

log  r=  I.I  1902 

^''-  I  ^^''^  -s{z;)-SjV)\  =  9-^0704 
log  sin  2co  =z  9.66331 

2C0  =  27°  25' 30'' 


CO 


=  13°  42' 45' 


I20  EXTERIOR    BALLISTICS. 

The  value  of  o;  computed  by  the  more  exact  foi  inula 

tan  CO  —  — - — ^—  \  I  {v) -^-^ —~j^  \ 

ig  2  cos'  (f  I     ^  ^        ^  (v)  —  ^  {V)  ) 

..==13°  21^  30'' 

differing  by  21'  from  the  less  approximate  value. 
We  have  found  above 

z;=  873.8  f.  s. 
but  this  is  only  an  approximation.     To  determine  its  true 
value,  that  is,  i^s  true  value  so  far  as  the  formulce  are  eonee?ned, 
we  should  have 

cos       10°  „^  r 

V  —  873.8 5 — -, — 7,  =  884.45  f .  s. 

'^     cos  13°  21' 30' 

differing  from  the  approximate  value  by  about  10  feet. 

Example  2. — "A  6-inch  projectile  leaves  the  gun  at  an 
angle  of  departure  of  4°,  with  an  initial  velocity  of  2100  f.  s. ; 
7e^=  64  lbs.,  </=  6  inches.  Find  the  range  in  horizontal  plane 
through  the  muzzle  of  the  gun,  and  time  of  flight."  (''  Ex- 
terior Ballistics,"  by  Lieutenants  Meigs  and  Ingersoll, 
U.S.N.) 

We  have  (Table  I.) 
5(F)  =  2024.8;^(F)  =  20.57;  /(F)  =  o.02246;  r(F)  =  o.838 

Takin^:  <:=i  i,  we  have 

Next  we  have  3^ 

AM^.20-57  ^  36  ,i„  go  +  /  (F)  ^  0..0074 
5  {v)  —  2024.8       64  \       \    J 

from  which  equation  we  readily  find 

V  =  993.77  f-  s. 
.' .  S  {v)  =  7801.8,  and  T  {v)=z  5.051 
X—  C  [7801.8  —  2024.8]  =  10270  ft. 

Problem  2. — Given  the  angle  of  fall  a7id  terminal  velocity,  to 
determine  the  initial  velocity,  angle  of  projection,  range,  and  time 
of  flight. 


'i 


EXTERIOR   BALLISTICS.  12 


We  have  [equation  (87)] 


A(v)~A{V)  _  .  /  X       sin  2co 
S  {v)  -  S{V)  -     ^""^  C~~ 

from  which  to  calculate  Fby  double  position. 

We  may  also  determine  V  by  the  equation  (see  Prob- 
lem i) 

r(F)  =  2  r(zO-  T{z') 

v^  being  found  by  the  equation 

/  (vo)  =  /  {v) ^— 

derived  from  (82). 

Having  found  F  by  either  method,  <p,  X,  and  Tare  com- 
puted by  the  equations 

^\A(v)  —  A{V)         ,,,^J 

X^C\S{v)-S{V)-] 

r=-^i  T{v)  —  T{V)\ 
COS  (p  {  ) 

Example  i.^Given  </=4.5  inches;  7e'=  35  lbs. ;  w=:  15°, 
and  z/=  772.74  f.  s. ;  to  determine  ^,  X,  and  T. 

It  will  be  found  that  we  have  the  following  equation  from 
which  to  find  V : 

2058.17  — ^(F) 

> -r -F777V  =  0.26807 

11633.6  —  ^  (F)  ^ 

For  the  first  trial  assume  V ^z  1500,  and,  substituting  in 
the  first  member  of  the  above  equation,  it  reduces  it  to 
0.26691,  which  is  too  small  by  0.00116  =  ^j.  Next  make 
F=  1480,  and  we  shall  find  that  the  first  member  now  be- 
comes too  great  by  0.00140  =:  e^;  then 
256  :  20  :  :  116  :  9.1 
The  correct  value  of  Fis  therefore  1500  —  9,1  =:  1490.9  f.  s., 
from  which  are  easily  found 

^  =  9°  51^  X=  12440  ft. ;    T=i2".72. 

Example  2. — "  In  attacking  a  place  with  curved  fire  it 
was  required  to  drop  shell  into  the  place  with  an  angle  of 


122  EXTERIOR   BALLISTICS. 

descent  of  12°,  and  terminal  velocity  of  600  f.  s.,  using  the 
8-inch  howitzer  and  a  projectile  of  180  lbs.;  find  the  requi- 
site position  of  the  battery,  and  the  requisite  elevation  and 
charg-e  of  powder."'^ 

Here   <3f=8   inches;    zv  =:  iSo   lbs.;    7^=600  f.   s.,    and 
co^  12°;  to  find  X,  V,  and  (p.     We  have 

log  sin  2C0  =  9.60931 
log  (7  =  0.44909 

log  0.14462  =  9.16022 
/  (7;)  =  1. 15929 

I{v^  =  1. 01467  v^  =  630.85  f.  s. 

whence  we  find 

T{V)  =  2X  14-396—  15779  =  13-012 

F:=  665.1    f.  S. 

5  (v)  =  15926.6 
5(F)  3^  141 78.9 

log  1747.7  =  3-24247 

log  A"=:  3.69156 

X=49i5  ft.  =  1638  yds. 

I{z'o)=    1. 01  467 

/(F)  =  0.87708 

log  0.13759  =  9.13859 
log  sin  2(p  =  9.58768 

2<p  =  22°  46'  ^  =   11°  23' 

Problem  3. — Given  the  range  and  initial  velocity,  to  deter- 
mine the  other  elements  of  the  trajectory. 

This  is  by  far  the  most  important  of  the  ballistic  prob- 
lems, and  it  happens,  fortunately,  to  be  one  of  those  most 
easily  solved  by  Siacci's  formulae. 

For  the  terminal  velocity  we  have 

*  Prof.  A.  G.  Greenhill  in  "  Proceedings  Royal  Artillery  Institution,"  No.  2,  vol.  xiii.  page  79. 


EXTERIOR   BALLISTICS. 


123 


and  then,  with  Fand  v  known,  all  the  other  elements  can  be 
computed  by  formulcE  already  considered. 

Example  i.— Find  the  elevation  required  for  a  range  of 
2000  yards  with  the  i6-pdr.  M.  L.  R.  i^im,  the  muzzle  velo- 
city being  1355  f.  s. ;  find  also  the  time  of  flight  and  angle 
of  descent. 

Here  <^r=:  3.6;  7(y  =  16;  log  (7  =  0.09152  ;  F=  1355,  and 
X  =  6000. 

Answer : 


4°  41 


T  =  5^91 
0^  =6°  13^ 
Example  2. — Compnte  a  range  table  for  the  Z-inch  rifle  {con- 
verted), up  to  15000  ft. 

We  have  for  chilled  shot,  7£^  =  183  lbs.;  <^=:  8  in.  (whence 
log  (7  =  0.45627),  and  V—  1404  f.  s.  First  take  from  Table 
I.  the  following  numbers,  which  are  to  be  used  in  all  the 
calculations : 

5  (r)  =  4858.5,  y^(F)r=  162.23,  /(F)  =  0.08595,  r(F)=:  2.500 

The  remainder  of  the  work  niay  be  concisely  tabulated 
as  follows : 


X 

ft. 

X 

c 

S{v) 

v 

A{v) 

/(v) 

T(v) 

1500 

524-59 

5383-1 

1303.0 

212.04 

0. 10442 

2.884 

3000 

1049 

2 

5907.7 

I2I2.8 

272.28 

.12579 

3 

305 

4500 

1573 

8 

6432.3 

II34-3 

344 -^o 

.15038 

3 

753 

6000 

2098 

4 

6956.9 

I  69.2 

430.79 

.17826 

4 

230 

7500 

2622 

9 

7481.4 

1019.2 

532.14 

. 20929 

4 

732 

9000 

3147 

5 

8006.0 

978.8 

650.68 

.24314 

5 

257 

10500 

3672 

I 

8530.6 

942.5 

787.72 

-27973 

5 

804 

12000 

4196 

7 

9055.2 

908.8 

944.68 

•31914 

6 

371 

13500 

4721 

3 

9579.8 

877-4 

1123.07 

.36148 

6 

959 

15000 

5245-9 

10104.4 

848.1 

1324.47 

. 40684 

7.567 

The  numbers  in  the  first  column  are  the  ranges  for  which 
the  elements  of  the  trajectory  are  to  be  computed.  The 
numbers  in  the  second  column  are  simple  multiples  of  the 
first  number  in  the  column.     Adding  S  {V)  to  the  numbers 


124 


EXTERIOR    BALLISTICS. 


in  the  second  column  ^ives  those  in  the  third  column,  and 
with  these  we  take  from  Table  1.  the  values  of  v,  and  at  the 
same  time  those  of  A  {^v),  I  {v),  and  T  {v). 

The  time  of  fliglit,  angle  of  departure,  and  angle  of  fall 
are  then  computed  by  the  following  formulas: 

'T=  — ^  \  T{v)-  T{V)\ 


cos  (p 


and 


sm  2lp: 


tan  Col  = 


ciiM^Am-ii^V) 


S{v)-S{V) 
C        (  ,  ,  ,       A  (v] 


-,{n^) 


A{V)) 


2  cos'  (p  {     ^  '        S  (v)  —  S  {V)  ) 
Lastly,  the  values  of  v,  tabulated  above,  a?-e  to  be  multi- 
plied by  cos  (f  sec  &->  to  obtain  the  correct  striking  velocities. 
In  our  example  the  results  are  as  follows: 


yds 

<!> 

a, 

T 

500 

o°44' 

o°47' 

1303 

I^IO 

1000 

i°33' 

i°43' 

I2I3 

2^30 

1500 

2°  2/ 

2°  50^ 

II35 

3''-59 

2000 

3°  2/ 

4°  08' 

1070 

4^96 

2500 

4°  32' 

5°  38' 

I02I 

6^40 

3000 

5°  43' 

7°  14' 

982 

7^92 

3500 

6°  59' 

9°  01^ 

947 

9^52 

4000 

8°  21^ 

10°  58^ 

916 

11^19 

4500 

9°  49' 

13°  06' 

888 

12^94 

5000 

11°  24' 

15°  25' 

862 

14^78 

By  interpolation,  using  first  and  second  differences,  the 
interval  between  successive  values  of  the  argument  {X)  may 
be  reduced  from  500  yards  to  100  yards. 

Example  3. — Given  d  —  20  93  cm. ;  if>  =  140  kg. ;  V  =  521 
m.  s. ;  d^  =  1.206;  d=  1.233  ;  X=4097  m.;  angle  o(  jump  =  8'; 
required  the  angle  of  elevation  ==  ^  —  8',  the  angle  of  fall, 
the  striking  velocity,  and  the  time  of  flight.^ 

Making  the  ballistic  coefficient  {c)  =0.907,  we  have  for 

*  '■  Ballistische  Formeln-von  Mayevski  nach  Siacci.  Fur  Elevationen  unter  15  Grad,"  Essen, 
Fried.  Krupp'sche  Buchdruckerei,  1883,  page  22.  Also  quoted  by  Siacci  in  "  Rivista  di  Artiglieria 
e  Genio,"  vol.  ii.  page  414,  who  solves  the  example,  using  Mayevski's  table. 


EXTERIOR   BALLISTrcS. 


125 


computing  C  in  English   units,  when  <^  is  expressed  in  centi- 
metres and  w  in  kil(3grarames,  the  following  expression  : 

C-[i..953743]f  ^ 

The  following  are  the  results  obtained  by  experiment, 
by  Mayevski's  calculations,  by  Siacci's  calculations,  and  by 
Table  I.  of  this  work : 


T 

Angle  of 
Elevation. 

Angle  of 
Fall. 

Striking  Velocity, 
f.  s. 

By  experiment 
Mayevski... 

Siacci 

Table  I 

9"-7 
9".6 

9".675 

9".66 

5°  30' 
5°  32' 

5°  31' 
5°  29'  30" 

7°  16' 

I  176 
I  169 

Example  4. — Given  ^=24  cm.;  7e/  =  2i5  kg.;  F=  529 
m.  s.  =  1735.6  f.  s. ;  required  the  angle  of  departure  for  each 
of  the  horizontal  ranges  contained  in  the  first  column  of  the 
followintr  table : 


Horizontal 

Range. 

in 

5/ 
J 

Computed  by 
Table  I. 

Observed 
value  of 

Values  of  <f>  computed  by 

Mayevski's 
Table. 

Hojel's 
Table. 

2026 

0.9569 

2°.;' 

2°  .9' 

2°   18' 

2°  14' 

3000 

0.9407 

3°  36' 

3°  41' 

3°  37^ 

3°  35' 

4000 

0.9756 

5°    5' 

5°  10' 

5°    6' 

5°    5' 

5964 

0.9560 

8"  41' 

8°  35' 

8°  44^ 

8°  44' 

7600 

0.9461 

12°  31' 

12°    5' 

12°  31' 

12°  32' 

The  data  in  the  first,  second,  and  fourth  columns  are 
taken  from  Krupp's  Bulletin,  No.  56  (February,  1885),  page 
4.  The  values  of  <p  in  the  third  column  were  computed  by 
Siacci's  method,  using  Table  I.  of  this  work.  In  the  last 
two  columns  are  given  the  values  of  ^  computed  by  Siacci's 
method  with  Mayevski's  and  Hojel's  tables  respectively. 

Problem  4. —  With  a  given  initial  velocity^  required  the  angle 
16 


126  EXTERIOR   BALLISTICS. 

of  projection  necessary  to  cause  a  projectile  to  pass  through  a 
given  point. 

Let  X  and  y  be  the  co-ordinates  of  the  given  point.     Then 
from  (83)  and  (84)  we  have 

and  ^ 

Example. — An  8-inch    service   projectile  is   fired  with  an 
initial  velocity  of  1404  f.  s.  from  a  point  33  feet  above  the 
water;   find  the  necessary   angle  of  projection    to   attain  a 
range  on  the  water  of  3000  yards. 
Here  <^=:  8,  ze/  =  180,  F=  1404,  x  =  9000  ft.,  and  j/=  —  33  ft- 

We  have  ^ 

^  (^^)  =  I^  ^  9000  +  4858.5  =  8058.5 

•••  ^  =  975-07 
In  calculating  tan  ^  we  will,  at  first,  omit  the  factor  cos"  (p 
in  the  second  member. 

33      ,    180(663.56—162.23  „        I 

=  —  0.00367  -|-  0.09945  =  0.09578 

Therefore  the  approximate  value  of  ^  is  5°  28'.     Complet- 
ing the  calculation  by  introducing  cos""  ip   we  have 

?  =  5°3i' 
which  needs  no  further  correction. 

Problem   5. — Given  the  initial  and  terminal  velocities,  to 
calculate  the  trajectory. 

For  the  solution  of  this  problem  we  have  the  following 
equations:  ^  A  {v)  -  A  {V)       ,,„,! 

^  \   ^'>     s{v)-s(y)S 


sm  2(« 


X=C[5(^)-5(F)] 

7-=-^  I  T{v)-  T{V)\ 
cos  <f  \        '  S 


EXTERIOR   BALLISTICS.  I27 

Example. — In  experimenting-  with  the  15-inch  S.  B.  gun, 
it  is  desired  to  place  a  target  at  such  a  distance  from  the 
gun  that  the  projectile  (solid  shot  weighing  450  lbs.)  shall 
have  a  velocity  of  1000  f.  s.  when  it  reaches  the  target,  and 
this  without  diminishing  the  muzzle  velocity,  which  is  1534 
f.  s.  What  is  the  required  distance  and  the  angle  of  pro- 
jection ? 

We  readily  find,  using  Table  II., 

and  ^  =  2°  33' 

X=4678  ft. 

CORRECTION   FOR  VARIATION  IN   THE    DENSITY    OF   THE   AIR. 

The  ballistic  coefficient  (Q  is  determined  by  the  equation 

r-  ^  h. 

cd'    d 
in  which  d^  is  the  adopted  standard  density  of  the  air,  and  d 
the  density  at  the  time  of  firing. 

In  computing  Tables  I.  and  II.  the  value  of  d^  was  taken 
as  the  weight,  in  grains,  of  a  cubic  foot  of  air  at  a  tempera- 
ture of  62°  F.  and  a  pressure  of  30  inches  of  mercury.  Ac- 
cording to  Bashforth  we  have 

'^z  =  534-22  grs. 

For  any  other  temperature  (/),  and  barometric  pressure 
{b)j  we  may  determine  the  value  of  d  near  enough  for  most 
practical  purposes  by  the  following  simple  equation: 
^_       20.212  b 
~  I  -f  .002178  t 

Correction  for  Altitude. — When  a  projectile  is  fired 
at  such  an  angle  of  projection  as  to  reach  a  great  altitude  in 
its  flight,  the  value  of  o,  determined  as  above,  will  be  too 
great.  We  may  calculate  0  approximately,  in  this  case,  as 
follows : 

If  o'  is  the  density  of  the  air  at  the  height  y  above  the 
surface  of  the  earth,  we  shall  have 

d'^de-'x 


128 


EXTERIOR   BALLISTICS. 


where  ?.  is  the  height  of  a  homogeneous  atmosphere  of  the 

density  <5,  which  would  exert  a  pressure  equal  to  that  of  the 

actual  atmosphere.'^' 

d  o       ^ 

The  factor  -—  becomes,  therefore,  ~  e^;   and  (7  must  be 
o  o 

multiplied  b}^  this  if  we  wish  to  take  into  account  the  dimi- 
nution of  density  due  to  the  height  of  the  projectile,  taking 
for  J  a  mean  value  for  the  arc  of  the  trajectory  which  we  are 
computing. 

y 

The  following  table  gives  the  values  of  ^-a.  for  every  lOO 
feet  from  j  =  o  to  /=  10,000  feet.  In  the  computation  ?. 
was  assumed  to  be  27800  feet,  which  is  its  approximate 
value  for  a  temperature  of  15°  C.  and  barometer  at  o"'.75. 
The  table  is  substantially  the  same  as  that  given  by  Bash- 
forth  {"  Motion  of  Projectiles,"  page  103),  but  in  a  moie  con- 
venient form. 


y 

0 

100 

200 

300 

400 

500 

6qo 

700 

800 

900 

0 

I. 0000 

0036 

0072 

0108 

0145 

0181 

0218 

0255 

0292 

0329 

1000 

1,0366 

0403 

0441 

0479 

0516 

0554 

0592 

0631 

0669 

'0707 

2000 

1.0746 

0785 

0824 

0863 

0902 

0941 

0981 

1020 

1060 

1 100 

3000 

I. I 140 

1180 

1220 

1260 

1 301 

1 34 1 

1382 

1423 

1464 

1506 

4000 

I -1547 

1589 

1630 

1672 

1714 

1756 

1799 

1841 

1884 

1927 

5000 

I. 1970 

2013 

2057 

2100 

2144 

2187 

2231 

2276 

2320 

2364 

6000 

1.2409 

2-154 

2499 

2544 

2589 

2634 

2679 

2725 

2771 

2817 

7000 

1.2863 

2909 

2956 

3003 

3049 

3096 

3H4 

3191 

3239 

3286 

8000 

I  3334 

3382 

3431 

3479 

3528 

3576 

3625 

3675 

3724 

3773 

9000 

1.3823 

3873 

3923 

3973 

4023 

4074 

4125 

4176 

4227 

4278 

*  Chauvenet's  "  Practical  Astronomy,"  vol.  i.  page  138. 


BALLISTIC  TABLES. 


The  term  ''Ballistic  Table"  was  applied  by  Siacci  to 
tlie  tabulated  values  of  the  funclions  S{v),  A  {v),  I{v),  and 
7\v).  Table  L  g-ives  the  values  of  these  functions  for  ob- 
long projectiles  having  ogival  heads  struck  with  radii  of  i| 
calibers.  It  is  based  upon  the  experiments  of  Bashforth, 
and  was  calculated  by  the  formulas  developed  in  the  preced- 
ing pages. 

The  table  extends  from  z/=28oo  to  ^  =  400,  which  limits 
are  extensive  enough  for  the  solution  of  nearly  all  practical 
problems  of  exterior  ballistics.  It  may  occasionally  happen 
in  mortar  practice  that  the  horizontal  velocity  {v  cos  <f)  may 
be  less  than  400  (as  in  problem  4,  Chapter  V.)  In  such 
cases  we  may  employ  the  formulas  by  which  this  part  of  the 
table  was  computed,  viz.: 

5  (v)  =  124466.4  -  [4.59i833(>]  log  ^ 

A  (v)  =  [9.6655206]  -^  -f  [4.1438598]  log  V  -  45916.40 

/(^)  =  [5.7369333]  ^  -  0.356474 
T{v)  =  [4.2296173]  ^  -  12.4999 

Example  i.— Let  (^=8  in.,  w  =  180  lbs.,  F=  700  f.  s.,  and 
^  =  60°.     Find  V  when  ^  =  —  60°. 
We  have  from  (33) 

and   U  ^=.  joo  cos  60°  =  350,   which  is  below   the   limit  of 


2  BALLISTIC   TABLES. 

the  table.     The  operation   may   be    concisely  arranged  as 
follows : 

const.  log=::  57369333 
2  log  f/=  5.0881360 

0.6487973  =  log  445448 
(60)  =  2.39053 

log  4  (60°)  =  0.9805542 
log  C=  0.4490925 

0.5314617  =  log  3.39987 

0.895 1103  =  log  7-85435 

2)4.8418230 

2  42091  15  =:  log  263.6 
.  • .  7/ =:  263.6  X  2  =  527.2  f.  S. 

Example  2. — Given  5  {v)  =  25496.8,  to  find  v. 
We  proceed  as  follows: 

1 24466.4 
25496.8 

log  98969.6  =  4.9954886 
const,  log  =  4.5918330 

log  (log  z/)  =  0.4036556 
.-.  log  7;=2.533i2 
£^=341.3 

Table  II.  is  the  ballistic  table  for  spherical  projectiles, 
and  extends  from  z^=  2000  to  ^^  =  450.  It  is  based  upon  the 
Russian  experiments  discussed  in  Chapter  II.,  and  is  be- 
lieved to  be  the  only  ballistic  table  for  spherical  projectiles 
yet  published. 

Table  III.  is  abridged  from  Didion's  "  Traite  de  Bal- 
istique." 

Forniulse  for  Interpolation. — To  find  the  value  of 
f{z^  when  V  lies  between  v^  and  v^,  two  consecutive  values 
of  V,  in  Tables  I.  and  II.     Let  v^  —  v^r=^  h.     Then,  if  d^  and  d^ 


BALLISTIC   TABLES. 


are  the  first  and  second  diflferences  of  the  function,  we  shall 
have,  since y(?7)  increases  while  v  decreases, 


2 


by  means  of  which  f{v)  can  be  computed.     Conversely,  if 
f{7>)  is  given,  and  our  object  is  to  find  v,  we  have 


7\  —  v\  d^ 
2 


In  using  this  last  formula,  first  compute  —^ —  by  omit- 


Ti 

ting  the  second  term  of  the  second  member  (which  is  usually 
very  small),  and  then  supply  this  term,  using  the  approxi- 
mate value  of-^-^^ —  already  found. 
Ii  ^ 

If  the  second  differences  are  too  small  to  be  taken  into 
account,  the  above  formulae  become 


/(z,)=/(t;,)  +  ^S-^rf, 


and 


which  expresses  the  ordinary  rules  of  proportional  parts. 

Example  i. — Find  from  Table  I.  S{v)  when  z/=  1432.6. 
We  have  v,  =  1435,  f{v^  =  4704.8,  h  —  5,  and  d,  =  24.6. 

.•.S{v)  =  4704.8  +  1435  -  1432.6  ^  ^^^^ ^ ^^j^^^ 

Example  2. — Given  A  (7/)  =  229.89,  to  find  v.     Here  7/^  = 
1274, /(7/,)  =  229.29,  </,=  1.25,  and /^=  2. 

2 

.  • .  7;  =  1 274 (229.89  —  229.29)  =  1 273.04 

1.25 

Example  3. — Find  from  Table  II.  A  {v)  when  77  =  517.8. 


4  BALLISTIC   TABLES. 

We    have   e^,  =  520,   ^(^0  =  3755-9.  >^^  =  5»  ^,  =  158.2,  and 
^,=  7.8. 

2.2  2.2  /  2.2X7.8 

.-.  ^  (^)  =  3755.9+ -X  158.2 ---(i--)^ 
=  3755-9  +  69-60  —  0.96  =  3824.5 
Example  4.— Find  from   Table  HI.  the  value  of  (^)  when 
^  =  54°  32'.     Here  ?^,  ==  54°  2o\  (^,)  =  17619 1»  h=z2o',d,z^ 
.02971,  d^  =  .00074. 

.-.  (^)z=  1.76191  +0.6X  0.02971  —0.6  X  0.4  X  0.00037 
=  1.76191  +0.01783  —0.00009=  1.77965 


TABLE  I. 


Ballistic  Tabic  for  Ogival-Hcaded  Projectiles. 


V 

6- (7') 

Diflf. 

A  iv) 

Diff. 

1 

Diff. 

T{v) 

Diff. 

2800 

2750 

2700 

j   000.0 

126.8 

[  256.0 

1268 
1292 
1315 

0.00 

0.07 
0.28 

7 
21 

36 

0.00000 
0.00106 
0.00218 

106 
112 

118 

0.000 
0.046 
0.093 

46 

47 
49 

2650 
2600 

2550 

387.5 

521.6 

658.3 

1341 
1367 

1393 

0.64 
1. 18 
1.89 

54 
71 
93 

0.00336 
0.00461 
0.00594 

125 

140 

0.142 
0.193 
0.246 

51 
53 
56 

2500 

2450 
2400 

797.6 
939.8 

1085.0 

1422 

1452 
1481 

2.82 
3.97 
5.37 

115 
140 
166 

0.00734 
0.00883 
0.01043 

149 
160 
169 

0.302 

0.359 
0.419 

57 
60 
62 

2350 

2300 
2250 

I233.I 

IJ84.5 

1539.2 

'514 
1547 
1582 

7.03 

9.00 

11.31 

197 
231 
266 

O.OI2I2 
0.01392 
0.01584 

180 
192 
205 

0.481 
0.546 
0.614 

65 
.  68 

72 

2200 
2190 
2180 

1697.4 

1729.5 
I76I.7 

321 
322 
323 

13.97 
14.55 
15.15 

58 
60 
62 

0.01789 
0.01832 
0.01876 

43 
44 
44 

0.686 
0.700 
0.715 

14 
^5 
15 

2170 

2160 

2150 

1794.0 
1826.5 
1859.2 

325 
327 
328 

15.77 
16.40 

17.05 

65 
67 

0.01920 
0.01964 
0.02010 

44 
46 
46 

0.730 

0.745 
0.760 

15 
15 

15 

2140 

2130 

2120 

1892.0 
1924.9 

1958.0 

329 
331 

17.72 
18.40 
19.10 

70 
73 

0.02056 
0.02102 
0.02149 

46 

47 
48 

0.775 
0.791 
0.806 

16 

15 
16 

2IIO 
2100 
2090 

I99I.3 

2024.8 
2058.4 

335 
336 
337 

19.83 

20.57 
21.33 

74 
76 

79 

0.02197 
0.02246 
0.02295 

49 
49 

50 

0.822 
0.838 
0.854 

16 
16 
16 

2080 
2070 
2060 

2092.1 
2126.0 
2I60.I 

339 
341 
343  > 

22.12 
22.92 
23.74 

80 
82 
85 

0.02345 
0.02396 
0.02447 

51 
51 

52 

0.870 
0.886 
0.903 

16 
17 
17 

2050 
2040 
2030 

2194.4 
2228.8 
2263.4 

344 
346 
348 

24.59 
25.46 

26.35 

[ 

87 
89 
91 

0.02499 
0.02552 
0.02606 

53 
54 
54 

0.920 
0.937 
0.954 

17 
17 
17 

2020 
2010 
2000 

2298.2 

2333.1 
2368.2 

349 
351 
353 

27.26 
28.20 
29.16 

94 
96 
98 

0.02660 
0.02715 
0.02772 

55 
57 
57  1 

0.971 
0.988 
1.005 

17 
17 
18 

TABLE  L— Continued. 


V 

S{v) 

Diff. 

A  {V) 

Diff. 

7(7') 

Diff. 

T{v) 

Diff. 

1990 

1980 

1970 

2403-5 
2439.0 
2474.6 

355 

^  356 

358 

30.14 
31-15 
32.19 

lOI 

104 
107 

0.02829 
0.02886 
0.02945 

57 
59 
60 

1.023 
1. 041 
1-059 

18 
18 
18 

i960 

1950 
1940 

2510.4 
2546.4 
2582.6 

360 
362 

363 

33-26 

34-35 
35-48 

109 
113 
115 

0.03005 
0.03066 
0.03127 

61 
61 
62 

1.077 
1.096 
1. 114 

19 

18 

19 

1930 

1920 
I9I0 

2618.9 
2655.5 
2692.2 

306 

367 

370 

36.63 
37-81 
39.02 

118 
121 
124 

0.03189 
0.03253 
0.03318 

64 
65 
65 

I-I33 
1. 152 
1. 171 

19 
19 

20 

1900 
1890 
1880 

2729.2 
2766.3 
2803.7 

371 
374 
375 

40.26 

41-53 
42.83 

127 
130 

0.03383 
0.03450 
0.03517 

67 
69 

1. 191 
1. 210 
1.230 

19 
20 
20 

1870 
i860 
1850 

2841.2 
2878.9 
2916.9 

377 
380 
382 

44.16 

1   45-53 
46.93 

137 
140 

143 

0.03586 
0.03656 

;  0.03727 

70 

71 

72 

1.250 
1.270 
1. 291 

20 
21 
20 

1840 

1830 

1820 

2955-1 
2993-4 
3032.0 

386 
388 

48.36 
49-83 
51-34 

147 
151 

155 

0.03799 
0.03872 
0.03946 

73  1 

74 

76 

1. 311 
1-332 
1-353 

21 
21 

22 

I8I0 

1800 
1790 

3070.8 
3109.8 
3149.0 

390 
392 
394 

52.89 

54-47 
56.09 

158 
162 
167 

'  0.04022 

0.04099 

10.04177 

77 
78 
80 

1-375 
1.396 
1.418 

21 

22 
22 

1780 
1770 
1760 

3188.4 
3228.0 
3267.9 

396 

399 
401 

1   57-76 

1   59-47 
61.21 

171 

174 
179 

i 
0.04257 

0.04338 
0.044.20 

81  1 
821 
84! 

1.440 
1.463 
1-485 

23 
22 

23 

1750 
1740 
1730 

3308.0 
3348.3 
3388.9 

403 
406 
409 

63.00 

64-83 
66.71 

183 
188 

193 

0.04504 
0.04589 
0.04676 

85  1 

87 

88! 

1.508 
1-531 

1-555 

23 
24 
23 

1720 
I7I0 
1700 

3429.8 
3470-8 
3512. 1 

410 
413 

415 

!    68.64 

:    70.61 
72.63 

1 

197 

202 
207 

0.04764 
0.04854 
0.04945 

90 

9r\ 

1-578 
1.602 
1.626 

24 
24 

25 

1690 

1680 

1670 

3553-6 
3595-4 
36374 

418 
420 
423 

1 
74-70 
76.83 
79.01 

213 
218 
223 

0.05038 
0.05133 
0.05229 

95 
96  1 
98  1 

1. 651 
1.676 
1. 701 

25 
25 
25 

1660 
1650 
1640 

3679-7 
3722.2 

3765-0 

425 
428 

430 

81.24 
83-52 
85.86 

228 

234 
241 

6 

0.05327 

0.05427 

,0.05529 

100 
102 

103  1 

1.726 

1-752 
1.778 

26 
26 
26 

TABLE  I.— Continued. 


3808.0 

3851-3 
3894.9 

3938.7 
3960.7 
3982.8 

4005.0 
4027.3 
4049.6 

4072.0 
4094.4 
4116.9 

4139-5 
4162.2 
4185.0 

4207.8 
4230.7 
4253-6 

4276.7 
4299.8 
4323-0 

4346.2 
4369.6 
4393-0 

4416.5 
4440.1 
4463-8 

4487-5 
4511-3 

I  4535-2 

4559-2 
4583.2 
4607.4 

4631.6 

4655-9 
4680.3 


Diff.  I 

I 

433  I 
436  I 
4381 

220  1 

221  j 

222  I 

223 
223 
224 

224 

225 
226 

227 
228 
228 

229 
229 
231 

231 
232 
232 

234 
234 

235 

236 
237 
237 

238 

239 

240 

240 
242 

242 

243 
244 
245 


A   {7') 


88.27 
90-73 
93-25 

95-84 
97.16 
98.49 

99.84 

IOI.2I 

102.60 
104.00 

105.42 
106.86 

108.32 

109.79 

111.29 

112.80 

114-33 

115.88 

117-45 

119.04 
120.65 


123-93 

125.60 

127.29 

129.01 

130.75 

132.50 
134.28 

136.09 

137.92 

139-77 
141.65 

T43-54 
T45-47 
147.42 


Diff. 


246 

252 
259 

132 
133 

135  I 

137 
139 
140 

142 
144 
146 

147 
150 
151 

153 

155 
157 

159 
i6i 
163 

165 
167 
169 

172 
174 

175 

178 
181 
183 

185 
188 


193 
195 
197 

7 


/{v) 


0.05632 
0.05738 
0.05845 

0.05955 

0.06010 
0.06066 

0.06123 
0.06180 
0.06238 

0.06296 

0.06355 

0.06414 
0.06474 

0.06534 
0.06595 

0.06657 
0.06719 
0,06782 

0.06846 
0.06910 

0.06975 

0.07040 
0.07106 

0.07173 

0.07241 

0.07309 
0.07378 

0.07447 
0.07517 
0.07588 

0.07660 

0.07732 
0.07805 

0.07879 

0.07954 

0.08029 


Diff. 


106 
107 


55 
56 

57 

57 
58 
58 

59 
59 
60 

60 
61 
62 

62 

63 
64 

64 
65 
65 

66 

67 
68 

68 
69 
69 

70 
71 

72 

72 
73 
74 

75 
75 
76 


T{v)' 

Diff. 

1.804 

1-831 
1.858 

27 
27 
27 

1.885 
1.899 

14 
14 

1.913 

14 

1.927 

14 

1. 941 

14 

1-955 

14 

1.969 
1.983 
1.998 

14 
15 
14 

2.012 

15 

2.027 

15 

2.042 

15 

2.057 

15 

2.072 
2.086 

14 
15 

2.101 

16 

2. 117 

15 

2.132 

15 

2.147 
2.162 

15 
16 

2.J78 

16 

2.194 

16 

2.210 

16 

2.226 

16 

2.242 

16 

2.258 

16 

2.274 

16 

2.290 

17 

2.307 

16 

2.323 

17 

2-340 

17 

2-357 

17 

2.374 

17 

TABLE  I. -Continued. 


V 

S{v) 

Diff. 

A  {j^ 

Diff. 

7(7.) 

Diff. 

r{v) 

Diff. 

1435 
1430 
1425 

4704.8 
1  4729-4 

i  4754-1 

246 

247 
247 

149-39 
151-39 

153-42 

200 

203 

205 

0.08105 
0.08182 
0.08260 

77 
78 
78 

2.391 
2.408 
2.425 

18 

1420 

I4I5 

I4I0 

1  4778.8 
1  4803.6 
,  4828.5 

248 
249 

250 

155-47 
T57-55 
159.66 

208 
211 
214 

0.08338 
0.08418 
0.08498 

80 
81 
81 

2.443 
2.460 

2.478 

1 

17 

18 
18 

1405 

1400 

1395. 

! 

J  4853-5 
i  4878.6 

49P3-8 

251 

252 

253 

1  161.80 
1  163.96 
j  166.15 

216 
219 
222 

0.08579 
0.08661 
0.08744 

82 

83 
84 

2.496 
2.514 
2^-532 

18 
18 
18 

1390 

1385 

1380 

4929-1 

4954-5 

j  4979-9 

254 
254 
256 

168.37 
170.62 
172.90 

225 
228 
231 

0.08828 
0.08913 
0.08999 

85 
86 

87 

2.550 
2.568 

2-587 

18 

19 
18 

1375 
1370 

'365 

5005.5 

■  5031-1 
5056.8 

256 

257 
258 

175.21 

177-55 
179.92 

234 
237 
241 

0.09086 
0.09173 
0.09262 

87 
89 
89 

2.605 
2.624 
2.643 

19 
19 
19 

1360 

1355 
1350 

5082.6 

!  5108.6 
5134.6 

260 
260 
261 

182.33 
184.76 
187.23 

243 
247 
250 

0.09351 
c. 09442 
0-09533 

91 
91 

93 

2.662 
2.681 

2.700 

19 

19 

.19 

1345 
1340 

1335 

5160.7 
5186.9 
5213-2 

262  i 
263' 

263  , 

1 

189.73 
192.27 
194.84 

254 

257 
260 

0.09626 
0.09719 
0.09813 

94 
94 
95 

2.719 

2-739 

2-758 

20 

•9 

20 

1330 
1325 

1320 

5239-5 
5265.8 
5292.0 

263! 

262  j 
106  , 

197.44 
200.06 
202.69 

262 
263 
107 

0.09908 
0.10004 
o.idioi 

96  1 

97 

39 

2-778 
2.798 
2.818 

20 

20 

8 

I3I8 
I3I6 
I3I4 

5302.6 

53^3-2 
5323-8 

106 : 

106 

107 

203.76 
204.84 
205.92 

108 
108 
109 

0.10140 
0.10179 
0.10219 

39  1 

40  1 

40' 

2.826 

2.834 
2.842 

8 
8 
8 

I3I2 
I3I0 

.1308 

5334-5 
5345-2 
5355-9 

107 

107 
108 

207.01 
208.11 
209.22 

I  10 
I  r  I 

III! 

1 

0.10259 
0.10299 
0.10339 

40 
40 
41 

2.850 
2.858 
2.866 

8 
8 
9 

1306 

1304 
1302 

5366.7 

5377-5 

108 
108 
109 

210.33 

211.45 

•212.58 

1 
112 

113 
114 

0.10380 
0.10421 
0.10462 

41 
41 
41 

2.875 
2.883 
2.892 

8 

9 
8 

1300 
1298 
1296 

5399-2 
5410.1 

5421.0 

109 
109 

no 

213.72 
214.87 
216.02  1 

115 
115 

117 

0.10503 
0.10544 
0.10586 

41 
42 
42 

2.900 
2.908 
2.917 

8 

9 

8 

TABLE  I.— Continued. 


V 

Six,) 

1 

Diff. 

1 

A{v) 

Diff. 

I{v) 

Diff. 

I 

:  T{v) 

Diff. 

1294 

I  292 
1290 

5432.0 

5443-0 
5454.0 

no 
no 
III 

1 

1  217.19 

1  218.36 

!  219.54 

1 

117 
118 
119 

0.10628 
0.10670 

0.10713 

42 
43 
43 

1    2.925 

2.934 

1    2.942 

9 

8 
8 

1288 
1286 

T284 

5465.1 
5476.2 

5487.3 

III 
III 
112 

220.73 
221.93 
223.13 

120 
120 
122 

0.10756 
0.10799 
0.10842 

43 
43 
44 

!    2.950 

1    2.959 

2.968 

9 
9 
9 

1282 
1280 
1278 

549«-5 
5509-7 
5521.0 

112 
113 
113; 

224.35 
225.57 
226.80 

122 
123 
124 

0.10886 
0.10930 

0.10974 

44 
44 
45 

2.977 
2.985 
2.994 

8 

9 
9 

1276 

1274 
1272 

5532.3 
5543-6 
5554-9 

113  i 
113! 

114 

228.04 
229.29 
230.54 

125; 

125 

127  ! 

0.I10I9 
0.11064 
0.11109 

45 
45 
45 

3.003 
3.012 
3.021 

9 
9 
9 

1270 

1268 
1266 

i  5566.3 

5589-1 

114 
114 
115 

231.81 
234.37 

127 
129 
J29 

0.11154 
0.11200 
0.11246 

46 
46 
46 

3-030 

3-039 
3.048 

9 
9 
9 

1264 
J262 
1260 

5600.6 
5612.1 
5623.7 

115 
116 

116! 

1 

235-66 
236.97 
238.28 

13.1 1 

131 1 

132 1 

0.11292 
0.11338 
O.I  1385 

46 
47 

47 

3-057 
3.066 

3-075 

9 
9 
9 

1258 

1256 

•1254 

5635-3 
5647.0 
5658.6 

117 
116 
117 

239.60 
240.94 

242.28 

134: 
134  i 
136! 

0.11432 
O.II479 
0.11527 

47 
48 
48 

3.084 
3-094 
3-103 

10 

9 
10 

1252 
1250 
1248 

5670.3 
5682.1 

5693-9 

118 
118 
118 

1 

243-64 
245.00 
246.37 

136  1 

1371 
139' 

O.II575 
O.I1623 
0.11671 

48 
48 
49 

3-1^3 
3.122 

9 

9 

10 

1246 

1244 
1242 

5705-7 
5717-6 
5729-5 

119 
119 
119 

247.76 
249-15 

250-55 

139; 
140  i 
142 

O.II72O 
O.II769 
0.11819 

49 

50 

50 

3-141 
3-150 
3.160 

9 
10 

9 

1240 

1238 

1236 

5741.4 
5753-4 
5765.4 

120 
120 
121 

251.97 

253-39 

254.83 

142 

144  j 

144 

0.11869 
O.II919 
0.11969 

50 
50 
5^ 

3.169 

3-179 
3-189 

10 

10 

9 

1234 
1232 ; 
1230 

5777-5 
5789.6 
5801.7 

121 
121 
122 

256.27 

257.73 
259.20 

146! 

147  1 

148  ■ 

0.12020 
O.T2071 
0.12123 

51 

52 
52 

3-198 
3.208 
3.218 

10 
10 
10 

TABLE  L— Continued. 


V 

S{v) 

Diff.  j 

1228 
1226 

1224 

5813.9 
5826.1 

5838.4 

i 
122 

123 
123! 

1222 
1220 
I218 

5850.7 
5863.0 

5875-4 

123 
124 
124 

I216 
I  2  14 
I2I2 

5887.8 

5900.3 

1  5912.8 

125  i 

125  1 
125 

I2IO 
1208 
1206 

5925.3 
5937-9 

1  5950.5 

126 
126 
127 

I  204 
1202 
1200 

5963.2 

5975-9 
5988.6 

127I 

127 

128 

1 

II98 
II96 
II94 

6001.4 
6014.2 
6027.1 

128 
129 
129 

1 

II92 
I  1 90 

I188 

6040.0 
6053.0 
6066.0 

130 
130 
131 

I  186 
I  184 
I182 

6079.1 
6092.2 
6105.3 

131 
131 

132  ! 

1 

I180 
II78 
II76 

6118.5 

6131-7 
6145.0 

132 

II74 
II72 
II70 

6158.3 
6171.7 
6185. I 

134 
134! 

135 ! 

I168 
I166 
1 164 

6198.6 
6212. 1 
6225.6 

135 
135 
136' 

A{v) 


260.68 
262.17 
263.67 

265.18 
266.71 
268.24 

269.79 

271-35 
272.92 

274.51 
276.11 

277.72 

279.34 
280.97 
282.62 

284.28 

285.95 
287.63 

289.33 
291.04 
292.76 

294.50 
296.25 
298.02 

299.80 

301.59 
303-40 

305.22 
307.06 
308.91 

310.77 
312.65 

314-55 


Diff. 

i 

149 

150 
151 

I{v) 

Diff. 

r(z/) 

O.I2I75 

0.12227 
0.12280 

52 
53 
53 

! 

3.228 
3-238 
3-M8 

'53 
153 

155 

0.12333 
0.12386 
0.12439 

53 
53 
54 

3-258 
3.268 
3-278 

156 

\    157 

159 

0.12493 

0.12547 

O.T2602 

54 
55 

55 

3.288 

1   3-299 

3-309 

1  160 
i  161 
1  162 

0.12657 

O.I  27  I  2 
0.12768 

55 
56 
56 

1   3-319 

i   3-329 

3-340 

163 

165 
166 

0.12824 
0.12881 
0.12938 

57 
57 
57 

3-350 
3-361 
3-371 

167 
168 
170 

0.12995 

O.T3053 
O.I3III 

58 
58 
58 

3-382 
3-393 
3.404 

171 

172 
1.74 

O.I3169 
0.13228 
0.13287 

59 
59 
60 

3.415 
3.426 

3.437 

175 
177 
178 

0.13347 
0.13407 

0.13467 

60 
60 
61 

3.448 
3.459 
3.470 

179 
181 
182 

0.13528 

0.13589 
0.13651 

61 
62 
62 

3.481 
3.492 

3.504 

i 

184 

185 
186 

O.I3713 
0.13776 

0.13839 

63 
63 
63 

3.515 
3-527 

3-538 ! 

188 
190 
191 

0.13902 
0.13966 

O.T403O 

64 
64 

65 

3-550 
3-561 

3-573' 

Diff. 


TABLE  1.— Continued. 


S{v) 


Diff. 


162 
160 
159 

158 
157 
156 

1531 

152 
151 
150 

149 

148 

147 

146 

145 
144 

143 
142 
141 

140 

139 
138 

137 
136 
135 

134 
133 
132 

131 
130 
129 


6239.2 
6252.8 
6259.7 

6266.6 

6273.4 
6280.3 

6287.2 
6294.1 
6301.0 

6307.9 
6314.8 
6321.8 

6328.8 

6335-7 
6342.7 

63497 
6356.7 
63637 

6370.7 
6377.8 
6384.8 

6391.9 
6399.0 

6406.  T 

6413.2 
6420.3 
6427.4 

6434.6 
6441.7 
6448.9 

6456.1 

6463.3 
6470.4 


136 
69 
69 

68 
69 
69 

69 
69 
69 

69 

70 
70 

69 

70 
70 

70 
70 
70 

71 

70 

71 

71 
7T 
71 


72 

71 

72 
72 

72 

71 

72 


A   {7') 


316.46 

318.39 
31936 

320.34 
321.32 
322.30 

323.28 
324.27 
325.26 

326.26 

327.26 

328.27 

329.28 
330.29 
331-31 

33^-33 
333-3^ 
334-39 

335-43 
336.47 
337-51 

338.56 
339-61 
340.67 

341.73 
342.79 
343-^6 

344-94 
346.02 

347.10 

348.19 
349.28 

350.38 


)iff. 

1(7') 

1 

Diff. 

193 

97 
98 

i 
I 
0.14095 

O.I4160 

O.I4I92 

65 
33 

98 
98 
98 

0.14225 
0.14258 
O.I429I 

33 
33 

33 

99 
99 

1  0.14324 
1  0.14358 

34 
33  ! 

T(v) 


Diff. 


100 

lOI 
lOI 

lOI 

102 
102 

103 

103 
104 

104 
104 

105 
105 

106 
106 

106 

107 

108 

108 
108 

109 

109 
1 10 

109 

II 


-.--too-! 
10.14391 

0.14425 

0.14458 

\   0.14492 

i  0.14526 
1  0.14560 
1  0.14594 

i  O.T4628 

;  0.14662 
0.14697 

0.14731 
0.14766 
0.1 480 1 

0.14836 

0.14871 
i  0.14906 

0.14942 
0.14977 
0.15013 

0.15049 
0.15085 
0.15121 

^0.15157 
io.15193 
'  0.15229 

34 

33 
34 
34 

34 
34 
34 

34 

35 
34 

35 
35 
35 

35 
35 
36 

35 
36 
36 

36 
36 
36 

36 
36 


3-584 
3-596 
3.602 

3.608 
3.614 
3.62c 

3.626 
3-632 
3-^3^ 

3-644 
3-650 
3656 

3.662 
3.668 
3-674 

3-680 
3.686 
3-693 

3-699 
3-705 
3. 711 

3-717 
3-723 
3-730 

3-736 

3-742 
3-748 

3-755 
3-761 
3-767 

3-774 
3-780 
3.786 


TABLE  I.— Continued. 


V 

S{v) 

Diff. 

Ah) 

Diff. 

I{z) 

Diff. 

T   (7') 

Diff. 

II28 
1.27 

II26 

6477-6 
6484.8 
6492.1 

72 

73 
72 

351  47 
352-57 
353-68 

no 
III 
111 

0.15265 

0.15302 
0.15338 

37 
36 
37 

3-793 

3-799 
3.806 

6 

7 
6 

II25 

II24 

6499.3 
6506.6 

73 
73 

354-79 
355-90 

II I  j 
113 

0.15375 
O.I54I2 

^7 
37 

3.812 
3-818 

6 
7 

II23 

65139 

73 

357-03 

113  1 

0.15449 

38  1 

3-825 

6 

II22  i 
II2I  1 
II20 

6521.2 
6528.6 
6536.0 

74 
74 
74 

358.16 
35930 
36045 

114 
115 
115 

0.15487 

0.15524 
0  15562 

37  1 
38; 

38  1 

3.831 
3-838 
3-844 

7 
6 

7 

III9 
II18 
I  I  17 

6543-4 
6550-8 
6558-3 

74 

75 
75 

361.60 
362.76 
36392 

116 
116 
117 

0.15600 

0.15638 
0.15676 

38 
38 
39 

3-851 
3.858 
3.864 

7 
6 

7 

II16 
11  14 

6565.8 

^573-3 
6580.8 

75 
75 
76 

365-09 
366.28 

367-47 

119 
119 

120 

0.15715 
0.15754 
0.15793 

39 
39 
39 

1   3.871 
3.878 
3-885 

7 
7 
7 

III3 
II12 

6588.4 
6596.0 

76 

77 

368.67 
369.88 

121 
121 

0.15832 

0.15872 

40 
40 

3.892 
3.898 

6 

7 

IIII 

6603.7 

77 

37109 

123 

0.15912 

40 

3905 

• 

7 

TIIO 

6611.4 

77 

372.32 

123 

0.15952 

41 

3912 

7 

1 109 
1108 

6619. 1 
6626.9 

78 
78 

373-55 
374-79 

124 
125 

O.T5993 
0.16033 

40 
41 

3.919 
3926 

1 

7 
7 

ITO7 
I  106 
1 105 

6634.7 
6642.5 
6650.3 

78 
78 
79 

376.04 

377-30 

i  37857 

126 
127 
128 

0.16074 
0.16115 
0.16157 

41 

42 

41 

3-933 
3940 
3-947 

7 
7 
8 

1 104 
I  103 
II02 

6658.2 
6666.2 
6674.1 

80 

79 

80 

379-85 
381.14 

382.44 

129 
130 
131 

0.16198 
0.16240 
0.16282 

42 
42 
43 

3-955 
3-962 

3-969 

7 
7 
7 

IIOI 
I  100 
1099 

6682.1 
6690.2 
6698.3 

81 
81 
81 

j  383-75 
38506 

T31 
132 

0.16325 
0.16367 
0.16410 

42 
43 
43 

3-976 
3-983 
3-991 

7 
8 

7 

1098 
1097 
1096 

6706.4 

6714-5 
6722.7 

81 
82 
83 

387-71 
389.06 

'  39041 

135 
135 
137 

0.16453 
0.16497 
0.16541 

44 
44 
44 

3-998 
4.006 

4013 

8 

7 
8 

TABLE  I.— Continued. 


V 

S{v) 

Diff. 

A^zi) 

Difif. 

nv) 

Diff. 

T{v) 

Diff. 

1094 1 
1093  i 

6731.0 
6739.2 
6747-5 

82 

83 

84 

391-78 
393-15 
394-53 

137 
138 

140 

0.16585 
0.16629 
0.16674 

44 
45 
45 

4.021 
4.029 
4-036 

8 

7 
8 

1092 
I09I 

1090 1 

6755-9 
6764.3 
6772.7 

84 
84 
85 

395-93 
397.34 
398.75 

141 
141 
142 

0.16719 
0.16764 
0.16810 

45 
46 
46 

4-044 
4.051 
4.059 

7 
8 
8 

1089 

1088 
1087 

6781.2 
6789.7 
6798.2 

85 
85 
86 

400.17 
401.60 
403-05 

143 
145 
145 

0.16856 
0.16902 
0.16948 

46. 
46 

47 

4.067 

4.075 
4.083 

8 
8 
8 

1086 
1085  j 

1084 

6806.8 
6815.4 
6824.1 

86  1 

87 

87 

1  404-50 
405.97 
407-45 

147 
148 
149 

0.16995 

0.17042 
0.17089 

47 
47 
48 

'  4.091 
4.098 
4.106 

7 
8 
8 

1083 

T082 
I08I 

6832.8 

6841.5 
6850.3 

87 
88 
88 

408.94 
410.44 
411.95 

150 
151 
152 

0.17137 

0.17185 
0.17233 

48 
48 

49 

4.114 
4.122 
4.130 

8 
8 
8 

1080 

1079 
1078 

6859.1 
6867.9 
6876.8 

88 
89 
90 

413-47 
415.00 

416.54 

153 
154 
156 

0.17282 

0.17331 
0.17380 

49 
49 
49 

4.138 
4.146 

4.155 

8 

9 
8 

1077 

1076 

1075 

6885.8 
6894.7 
6903.7 

89 
90 

91 

418.10 
419.66 
421.24 

156 
158 
159 

0.17429 

0.17479 
0.17529 

50 
50 
51 

4.163 
4.172 
4.180 

9 

8 

9 

1074 

ro73 
1072 

6912.8 
6921.9 
6931. 1 

91 
92 
92 

422.83 
424.44 
426.06 

161 
162 
163 

0.17580 
0.17631 
0.17682 

51 
51 
51 

4.189 

4.197 
4.206 

8 

9 

8 

1071 

1070 
1069 

6940.3 
6949.5 
6958.8 

92 
93 
93 

427.69 

429.33 
430.98 

164 

165 
166 

0.17733 
0.17785 

0.17837 

52 

52 
53 

4.214 
4.223 
4.232 

9 
9 
9 

1068 
1067 
1066 

6968.1 

6977-5 
6986.9 

94 
94 
94 

432.64 

434.32 
436.01 

168 
169 
171 

0.17890 

0.17943 
0.17996 

53 
53 
53 

4.241 

4.250 

i   4.259 

9 
9 
9 

1065 
1064 
1063 

6996.3 

7005.8 
7015-4 

95 
96 
96 

437.72 
439-44 
441.17 

172 
173 
175 

0.18049 
0.18103 
0.18158 

54 

55 
55 

4.268 

4.277 
.  4.286 

9 
9 
9 

^3 


TABLE  I.— Continued. 


V 

S{v) 

Diff. 

A  (v) 

Diff. 

I{v) 

Diff. 

1  T{v) 

1 

Diff. 

1062 
1061 
1060 

7025.0 
7034.6 
7044-3 

96 
97 
97 

442.92 
444-68 
446.45 

176 

177 
178 

0.18213 
0.18268 
0.18323 

55 
55 
56 

1 

1   4-295 

i   4-304 

4-313 

1 

1 

9 

9 

i    9 

1059 
1058 

1057 

7054.0 
7063.8 
7073.6 

98 
98 
99 

448.23 
450-03 
451.84 

180 
181 
182 

0.18379 

0-18435 
0.1 849 1 

56 
56 
57 

']       4-322 
4-332 
4.341 

10 
9 
9 

1056 

1055 
1054 

1  7083.5 
7093-4 
7103.4 

99 
100 
100 

453.66 

455.50 
457-36 

184 
186 

187 

0.18548 
0.18605 
0.18663 

57 
58 
58 

4-350 
4-360 
4-369 

10 
9 
9 

1053 

1052 
105 1 

7113-4 
7123.4 

7133-5 

100 

lOI 
I02 

459.23 
461.12 
463.02 

189 
190 
192 

O.18721 
0.18779 
0.18838 

58 
59 
59 

4.378 
4.387 
4.397 

9 
10 

9 

1050 
I049 
1048 

7143.7 

7153-9 
7164.1 

102 
I02 
103 

464-94 
466.87 
468.81 

193 
194 
196 

0.18897 
0.18956 
O.I  90 1 6 

59 
60 
61 

4.406 
4.416 
4.426 

10 
10 
10 

1047 
1046 
1045 

7174-4 
7184.7 

7195-I 

I03 
104 

105 

470-77 
472-74 
474-73 

197 
199 
201 

0.19077 
0.19138 
0.19199 

61 
61 
61 

4.436 
4.446 

4-455 

10 

9 
10 

1044 

1043 
1042 

7205.6 
7216.1 
7226.6 

105 

105 

106 

476.74 

478.77 
480.81 

203 
204 
206 

0.19260 
0.19322 
0.19385 

62 
63 

4-465 
4-475 
4-485 

10 
10 
10 

1041 
1040 
1039 

7237.2 
7247.9 
7258.6 

107 
107 
107 

482.87 

484.95 
487.04 

208 
209 
211 

0.19448 
0.19511 

0-19575 

^3 
64 
64 

4-495 
4-505 
4-516 

10 
II 
10 

1038 

1037 
1036 

7269.3 
7280.1 
7291.0 

108 

109 
109 

489-15 
491.28 

493.42 

213 
214 
216 

0-19639 
0.19703 
0.19768 

64 

65 
66 

4-526 

4.537 
4.547 

II 

10 
II 

1035 
1034 

7301.9 
7312.9 

73239 

no 
no 
III 

495.58 
497.76 
499-95 

218 
219 
222 

0.19834 
0.19900 
0.19966 

66 
66 
67 

4.558 
4-569 
4-579 

II 
10 
II 

1032 
1031 
1030 

7335.0 
7346.1 
7357.3 

III 
112 
112 

502.17 
504.40 
506.65  ' 

223 
225 
226  1 

0.20033 
0.20100 
0.20168 

67 
68 
68 

4-590 
4.600 
4.611 

10 
II 
II 

14 


TABLE  I.— Continued. 


V 

S{v) 

Diff. 

A{v) 

Diff. 

I{v) 

Diff. 

T{.v) 

Diff. 

1029 

1028 

1027 

7368.5 
7379-8 
739I-I 

113 
113 
114 

i 

508.91 
511.20 
513-50 

229 
230 

232 

1 

0.20236 
0.20305 
0.20374 

69 
69 
69 

4.622 
4-633 
4.645 

II 
12 
II 

1026 
1025 
1024 

7402.5 
7414.0 

7425-5 

115 
115 
116 

515-82 
518.17 

520.54 

235 
237 
238 

0.20443 
0.20513 
0.20584 

70 
71 
71 

4.656 
4.667 
4.678 

II 
II 
II 

1023 

1022 

I02I 

7437-1 
7448.7 
7460.4 

116 
117 
117 

522.92 

525-32 
527-75 

240, 

243 

245 

0.20655 
0.20726 
0.20798 

71 

72 

73 

4.689 
4.701 
4.712 

12 
II 
II 

1020 
IOI9 
IO18 

7472.1 
7483.9 
7495.7 

118 
118 
ii9| 

530.20 
532.66 
535-14 

! 
246 
248 
251 

0.20871 
0.20944 
O.21017 

73 
73 
74 

4.723 
4.735 
4-747 

12 
12 
12 

ICI7 
IO16 
IOI5 

7507.6 
7519-6 
7531-6 

120 
120 
121 

537-65 

540.17 
542.72 

252 
255 
258 

O.21091 
0.2 1 165 
0.21240 

1 

74 
75 
76 

4.759 
4.771 
4-782 

12 
II 
12 

IOI4 
IOI3 
IOI2 

7543-7 
7555-8 
7568.0 

121 
122 
123 

545-30 
547-89 

550.51 

259 
262 

265 

O.21316 
0.21392 
0.21468 

76 
76 
77 

4.794 
4.806 
4.818 

12 
12 
12 

lOII 
lOIO 
1009 

7580.3 
7592.6 
7605.0 

123 
124 
124 

553-i6 

555-82 
558-51 

266 
269 

272 

0.21545 
0.21623 
O.21701 

78 
78 
79 

4.830 
4.842 

4.855 

12 

13 

12 

1008 
1007 
1006 

7617.4 
7629.9 
7642.5 

125 
126 

1.6 

561.23 
563.96 
566.71 

273 

275 
278 

0.21780 
0.21859 
0.21939 

79 
80 
80 

4.867 
4.880 
4.892 

13 
12 

13 

1005 
1004 
1003 

7655-1 
7667.8 
7680.6 

127 
128 
128 

569-49 
572.29 

575.11 

280 
282 
285 

0.22019 
0.22100 
0.22182 

81 
82 
82 

4.905 
4.918 

4-930 

13 
12 

13 

1002 
lOOI 

1000 

7693-4 
7706.3 

7719-3 

129 
130 
131 

577-96 
580.83 

583-72 

287 
289 
292 

0.22264 
0.22347 
0.22430 

83 
83 
84 

4.943 
4.955 
4.968 

12 
13 
13 

999 
998 

997 

7732.4 
7745-6 
7758.8 

132 
132 
133 

586.64 

589-59 
592.56 

295 
297 
300 

0.22514 
0.22599 
0.22684 

85 
85 
86 

4.981 

4.995 
5.008 

14 
13 
14 

15 


TABLE  I.— Continued. 


V 

S\v) 

Diff. 

A{t^ 

Diff. 

/(zO 

Diff. 

T{v) 

Diff. 

996 

995 
994 

7772.1 

7785.4 
7798.7 

134 

595.56 

598.59 
601.65 

303 
306 

Z^9 

0.22770 
0.22857 
0.22944 

87 
87 
87 

5.022 

5.035 
5.048 

T3 
13 
14 

993 
992 
991 

7812. 1 

7825.5 
7839.0 

134 
135 
135 

604.74 
607.85 
610.99 

311 
314 
317 

0.23031 
0.23118 
0.23206 

87 
88 

89 

5.062 

5-075 
5.089 

13 
14 
13 

990 
989 
988 

7852.5 
7866.1 
7879.7 

136 
136 
137 

614.16 

617.33 
620.52 

317 

319 
321 

0.23295 
0.23384 
0.23474 

89 
90 
90 

5.102 
5. 116 
5-130 

14 
14 
14 

987 
986 
985 

7893.4 
7907.1 
7920.8 

137 
^37 
137 

623.73 
626.96 
630.21 

323 
325 
327 

0.23564 
0.23655 
0.23746 

91 
91 
91 

5-144 
5.158 
5-171 

14 
13 
14 

984 

983 
982 

7934-5 
7948.3 
7962.1 

138 
138 
138 

633-48 
636.77 
640.08 

329 

0.23837 
0.23929 
0.24021 

92 
92 
92 

5.185 
5.199 
5.213 

14 
14 
14 

981 
980 
979 

7975-9 
7989.8 
8003.7 

139 
139 
139 

643-41 
646.76 
650.12 

335 
339 

0.24T13 
0.24206 
0.24299 

93 
93 
93 

5.227 
5-241 

5-255 

14 
14 
15 

978 

977 
976 

8017.6 
8031.5 

8045.5 

139 
140 
140 

653-51 
656.92 
660.35 

341 
343 

345 

0.24392 
0.24486 
0.24580 

94 
94 
95 

5.270 
5.284 
5-299 

14 
15 
14 

975 
974 
973 

8059.5 
8073-5 
8087.6 

140 
141 
141 

663.80 
667.26 
670.75 

346 
349 

351 

0.24675 

0.24770 
0.24865 

95 
95 
96 

5-3^3 

5-327 
5.342 

14 
15 
14 

972 
971 
970 

8101.7 
8115.8 
8129.9 

141 
141 
142 

674.26 
677.80 
681.35 

354 

355 

357 

0.24961 

0.25057 
0.25154 

96 
97 
97 

5.356 
5.371 
5-385 

15 
14 
15 

969 
968 
967 

8144. 1 

8158.3 
8172.5 

142 
142 
143 

684.92 
688.51 
692.12 

359 
361 

363  1 

0.25251 
0.25348 
0.25446 

97 

98 
98 

5.400 

5.415 
5-429 

15 
14 
15 

966 

965 
964 

8i86.8 
8201. 1 
8215.4 

143 
143 
144 

695.75 
699.41 

703-09 

366 
368 

370 

0.25544 
0.25643 
0.25742 

99 
99 
99 

5-444 
5-459 
5-474 

15 

15 
,  15 

16 


TABLE  1.— Continued. 


V 

i 

Diff. 

A(v) 

Diff. 

/{v) 

Diff. 

T(v) 

Diff. 

963 

962 
961 

:   8229.8 
8244.2 
8258.6 

144 
144 
144 

706.79 
710.51 
714.26 

372 
375 

377 

0.25841 
0.25941 

0.26041 

100 
100 

lOI 

5-489 

5-503 

j   5-518 

1 

14 
15 

15 

960 

959 

958 

1 

8273.0 

\       8287.4 
i   8301.9 

144 
145 
145 

718.03 
721.81 

1  725.62 

378 
381 
384 

0.26142 
0.26243 
0.26344 

lOI 

101 

102 

5-533 
'   5-548 
1   5-564 

15 
16 

15 

957 
956 
955 

8316.4 

!   8331.0 

!  8345.6 

146 
146 
146 

729.46 

1  733.32 
737.20 

386 
3SS 
390 

0.26446 
0.26549 
0.26652 

103 
103 
103 

5-579 
5-594 
5.609 

15 
15 
16 

954 
953 
952 

i  8360.2 

8374.8 

i  8389.5 

146 
147 
J47 

741.10 

745-03 
748.98 

393 
395 
398 

0.26755 
0.26858 

0.26962 

103 

T04 

105 

5-625 
5-640 
5-655 

15 
15 
16 

951 
950 
949 

8404.2 

i  8419.0 

8433.8 

148 
T48 
148 

752.96 
756.96 
760.98 

400 
402 
404 

0,27067 

0.27172 
0.27277 

105 
105 

106 

5-671 
5.686 

5-702 

T5 
16 

948 

947 
946 

8448.6 
1  8463.4 
,  8478.3 

148 
149 
149 

765.02 
769.09 
773.18 

407 
409 
412 

0.27383 

0.27489 

0.27596 

106 

107 

107 

5-718 

■5-733 

5-749 

15 
16 
16 

945 
944 
943 

8493.2 
8508.1 

!  8523.1 

149 
150 

150 

777-30 

781.45 
785.62 

415 
417 
420 

0.27703 

0.278II 

0.27919 

108 
108 
108 

5-765 
5-781 
5-797 

16 
16 
15 

942 
941 
940 

1  8538.1 

\  8553.1 
\  8568.2 

1 

150 
151 
151 

789.82 
794.04 
798.29 

422 

425 
427 

0.28027 
0.28136 
0.28246 

109 
no 
no 

5-812 
5-828 
5-844 

16 
16 
16 

939 

938 
937 

8583.3 
8598.4 
8613.6 

151 

152 
152 

802.56 
806.85 
811. 17 

429 

432  1 
435 

0.28356 
0.28467 
0.28578 

III 
III 
III 

5.860 
5-877 
5-893 

17 
16 
16 

936 
935 
934 

8628.8 
8644.0 
8659.2 

152 
152 
153 

815-52 
819.89 
824.30 

437 
441 

443 

0.28689 
0.28801 
0.28913 

112 
112 
113 

5-909 
5-926 
5-942 

17 
16 
16 

933 
932 
931 

8674.5 
8689.8 
8705.2 

153 

^54 
154 

.  828.73 
837-67 

445 
449 
451 

0.29026 
0.29140 

0.29254 

114 
114 
114 

5-958 
5-974 
•5-991 

16 

17 
16 

T7 


TABLE  I.— Continued. 


V 

S{v) 

Diff. 

A{v) 

Diff. 

/(Z') 

Diff. 

Tiv) 

Diff. 

930 
929 
928 

8720.6 
8736.0 
8751.5 

154 
155 
155 

842.18 
846.71 

851.27 

453 
456 
459 

0.29368 
0.29483 

0.29598 

^15 
115 
116 

6.007 
6.024 
6.041 

17 
17 
16 

927 

926 

925 

8767.0 
8782.5 
8798.0 

155 
155 
156 

855.86 
860.48 
865.13 

462  1 

465 
468 

0.29714 

0.29830 

0.29947 

116 
117 

117 

6.057 

6.074 
6.091 

17 
17 
17 

924 
923 

922 

8813.6 
8829.2 
8844.9 

156 
157 
157 

869.81 
874-51 
879-25 

470 
474 
477 

0.30064 
0.30T82 
0  30300 

118 
118 
119 

6.108 
6.125 
6. 141 

17 
16 

17 

921 

920 

919 

8860.6 
8876.3 
8892.0 

157 
157 
158 

884.02 
888.81 

479  1 
4821 

485  j 

0.30419 
0.30538 

0.30658 

119 

120 
120 

6.158 

6.175 
6.192 

17 
17 
18 

918 
917 

916 

8907.8 
8923.7 
8939-5 

159 
158 
159 

898.48 
903.36 
908.27 

488 
491 
494 

0.30778 
0.30899 
0.31020 

121 
121 
122 

6.210 
6.227 
6.245 

17 
18 

17 

915 
914 

913 

8955-4 
8971-3 
8987-3 

159 
160 
160 

913.21 
918.18 
923.19 

497 
501 

503 

0.3II42 
0.31264 
0.31387 

122 
123 
124 

6.262 
6.279 
6.297 

17 
18 

17 

912 
911 

910 

9003-3 
9019.3 

9035-4 

160 
161 
161 

928.22 
933-28 
938.37 

506 
509 
513 

0.3I5II 
0.31635 
0.31760 

124 
125 
125 

6.314 
6.332 
6.349 

18 

17 
18 

909 
908 
907 

9051-5 
9067.6 
9083.8 

161 
162 
162 

943-50 
948.65 

953-84 

515 
519 

522 

0.31885 
0.320II 

0.32137 

126 
126 
127 

6.367 

6.385 
6.403 

18 
18 
18 

906 

905 
904 

9100.0 
9116.2 
9132.5 

162 

163 
163 

959.06 
964.31 
969.60 

525 
529 
532 

0.32264 

0.32392 

0.32520 

128 
128 
129 

6.421 

6.439 
6.457 

18 
18 
18 

903 

902 
901 

9148.8 
9165.2 
9181.6 

.64 

164 
164 

974-92 
980.27 

985-65 

535 
538 
541 

0.32649 
0.32778 

0.32908 

129 
130 
130 

6.475 
6.493 
6. 511 

18 
18 
18 

900 

899 
898 

9198.0 

9214-5 
9231.0 

165 
165 
165 

991.06 

996.51 
1001.99 

545 
548 

552 

0.33038 
0.33169 
0.33300 

131 
131 
132 

6.529 
6.548 
6.566 

19 

18 

19 

iS 


TABLE  I.— Continued. 


S(v) 


9247-5 
9264.1 
9280.7 

9297.3 
9314-0 
9330-7 

9347.5 
9364.3 
9381. 1 

9398.0 
9414.9 
9431-9 

9448.9 
9465.9 
9483.0 

9500.1 
9517.2 
9534.4 

9551.6 
9568.9 
9586.2 

9603.5 
9620.9 

9638.3 

9655.8 

9673.3 
9690.8 

9708.4 
9726.0 
9743.7 

9761.4 
9779.T 
9796.9 


Diff. 


166 
166 
166 

167 
167 
168 

168 
168 
169 

169 
170 

170 

170 
171 
171 

171 
172 
172 

173 
173 
173 

174 
174 

175 

175 
175 
176 

176 
177 
177 

177 
178 
178 


A{v) 


007.51 
013.06 
018.65 

024.27 
029.92 
035-61 

041.34 
047.10 
052.90 

058.73 
064.60 
070.52 

076.47 
082.45 
088.47 

094-53 
100.62 
106.75 

112.92 
119.13 
125-38 

131.67 

138.00 
144-37 

150.78 

157-23 
163.72 

170.25 
176.82 
183.44 

190.09 
196.79 
203.54 


Diff. 


555 
559 
562 

565 
569 
573 

576 
580 
583 

587 
592 
595 

598 
602 
606 

609 
613 
617 

621 
625 
629 

^33 
637 
641 

645 
649 
653 

657 
662 
665 

670 

675 
678 


I{v) 


0.33432 
0.33565 
0.33698 

0.33832 
0.33966 
0.34101 

0.34237 
0.34373 
0.34510 

0.34647 

0.34785 
0.34924 

0.35063 
0.35203 
0.35344 

0.35485 
0.35627 

0-35770 

0.35913 

0.36057 

0.36202 

0.36347 
0.36493 
0.36639 

0.36786 

0.36934 
0.37083 

0.37232 
0.37382 

0.37532 

0.37683 

0.37835 
0.37988 


Diff. 


133 
^33 
134 

134 
135 
136 

136 
137 
137 

138 
139 
139 

140 
141 
141 

142 
143 
143 

144 
145 
145 

146 

146 
147 

148 
149 
149 

150 
150 
151 

152 
^53 
153 


T{v) 


6.585 
6.603 
6.622 


6.640 
6.659 
6.677 

6.696 
6.714 
6.733 

6.753 
6.772 
6.791 

6.811 
6.830 
6.849 

6.868 
6.888 
6.907 

6.927 

6.947 
6.966 

6.986 
7.006 
7.026 

7.046 
7-065 
7.085 

7.105 
7.126 
7.146 

7.167 
7.187 
7.208 


Diff. 


19 


TABLE  I.— Continued. 


864 
863 
862 

861 
860 
859 

858 

857 
856 

855 
854 
853 

852 

851 
850 

849 
848 

847 

846 

845 
844 

843 
842 
841 

840 

839 
838 

837 
836 

835 
834 
832 


Siv) 


9814.7 
9832.6 
9850-5 


•4 
9886.4 
9904.4 

9922.5 
9940.6 
9958.7 

9976.9 

9995-2 
10013.5 

10031.8 
10050.2 
10068.6 

10087. 1 
10105.6 
10124.1 

10142.7 
10161.3 
10180.0 

10198.8 
10217.5 
T0236.3 

10255.2 
10274.1 
10293.0 

10312.0 
10331.0 
10350.1 

10369.2 
10388.4 
10407.6 


Diff. 


179 
179 

1791 

I 
180 
180 
181 

181 
181 
182 

183 
183 
183 

184 
184 

185 

185 
185 
186 

186 
187 
188 

187 
188 
189 

189 
189 
190 

190 

191 
191 

192 
192 
193 


A{v) 


210.32 
217.15 
224.02 

230.93 
237.89 
244.89 

251-94 
259.04 
266.18 

273.36 
280.59 
287.87 

295-19 
302.56 

309-98 

317-44 
324.96 

332.52 

340.13 
347.79 

355-50 

363-26 
371.07 

378.93 

386.84 
394.80 
402.82 

410.89 
419.01 

427.18 

435-41 
443.69 

452.02 


Diff. 


683 
687 
691 

696 

700 
705 

710 

714 
718 

723 
728 

732 

737 
742 
746 

752 
756 
761 

766 
771 
776 

781 
786 
791 

796 

802 
807 

812 
817 
823 

828 
^33 
839 


I{v) 


I0.38I4T 
:  0.38295 
;  0.38450 

0.38606 
0.38762 
0.38919 

0.39077 
0.39235 
0.39394 

0.39554 
0.39715 
0.39877 

0.40039 
0.40202 
0.40366 

0.40530 
0.40695 
0.40861 

0.41028 
0.41196 
0.41364 

0.41533 
0.41703 
0.41874 

0.42046 
0.42218 
0.42392 

0.42566 
0.42741 
0.42917 

0.43093 
0.43271 

0.43449 


Diff. 


154 
155 
156 

156 
157 
158 

158 

159 
160 

161 
162 
162 

163 
164 
164 

165 
166 
167 

168 
168 
169 

170 
171 
172 

172 
174 
174 

175 
176 
176 

178 
178 
180 


T(v) 


7.229 
7.249 

7.270 

7.290 

7-311 

7-332 

7-354 
7-375 
7-396 

7.418 

7-439 
7.460 

7.481 
7.503 

7-524 

7-546 
7-568 
7-590 

7.612 
7-635 
7-657 

7-679 
7.701 

7-723 

7-745 
7-768 

7-790 

7-813 
7.836 

7-858 

7.881 

7-904 
7.928 


TABLE  I.— Continued. 


S{v) 


Diff. 


0426.9 
0446.2 
0465.6 

[0485.0 
[0504.4 
0523-9 

0543-4 
0563.0 
0582.7 

[ 0602. 4 
0622,1 
[0641.9 

0661.7 
0681.6 
0701.6 

0721.6 
0741.6 
0761.7 

:o78i.8 
:o8o2.o 
:o822.2 

:o842.5 
0862.8 
:  0883. 2 

10903.6 
0924.1 
0944.6 

0965.2 
0985.8 
1006.5 

1027.2 
1048.0 
1068.8 


193 
T94 
194 

194 
195 
195 

196 
197 
197 

197 
198 
198 

199 

200 
200 

200 
201 
201 

202 
202 
203 

203 
204 
204 

205 
205 
206 

206 

207 
207 

208 
208 
209 


Aiv) 


460.41 
468.85 
477.35 

485.90 

494.51 
503-18 

511.90 
520.69 
529-52 

538.42 
547-38 
556.39 

565-47 
574.61 
583-80 

593-05 
602.37 
611.75 

621.20 
630.70 
640.27 

649.90 
659-60 
669.36 

679.19 
689.08 
699.04 

709.07 
719.16 
729.32 

739-55 
749-84 
760.21 


Diff. 


844 

850 
855 

861 
867 

872 

879 


896 
901 
908 

914 
919 
925 

932 
938 
945 

950 
957 
963 

970 
976 

983 

989 

996 

1003 

1009 
1016 
1023 

1029 
1037 
1043 


/{v) 


0.43629 
0.43809 
0.43990 

0.44172 
0.44354 
0.44538 

0.44722 
0.44908 

C.45094 

0.45282 
0.45470 
0.45659 

0.45849 

0.46040 
0.46231 

0.46424 
0.46618 
0.46812 

0.47008 

0.47205 

0.47402 

0.47601 
0.47800 
0.48001 

0.48202 
0.48404 

0.48608 
0.48812 

0.49018 

0.49225 
0.49432 

0.49641 

0.49850 


Diff. 


180 
181 
182 


182 
184 
184 

186 
186 


188 
189 
190 

191 
191 
T93 

194 
194 
196 

197 
197 
199 

199 
201 
201 

202 
204 
204 

206 

207 
207 

209 
209 
211 


T(v) 


7-95T 
7-974 
7-997 

8.021 
8.044 
8.068 

8.091 
8.115 
8.139 

8.163 
8.187 
8.211 

8.235 
8.259 
8.284 

8.308 
^'333 
8.357 

8.382 
8.407 
8.432 

8.457 
8.482 

8.507 

8.533 
8.558 
8.584 

8.610 

8-635 
8.661 

8.687 
8.713 
8.739 


Diff. 


TABLE  I.-rCONTINUED. 


Siv) 


798 

797 

796 

795 
794 
793 

792 
791 
790 

789 

788 
787 

786 
785 
784 

783 
782 

781 

780 

779 

778 

777 
776 

775 

774 
773 

772 

771 

770 
769 

768 
767 
766 


1089.7 
1 1 10. 7 
1131.7 

1152.7 
1173.8 
1195.0 

1216.2 

1237.5 
1258.8 

1280.3 
1301.8 
1323-4 

1345-0 
1366.6 

1388.2 

1409.8 
1431-5 
1453-3 

I475-0 
1496.8 
1518.6 

1540.4 
1562.2- 
1584.1 

1606.0 
1627.9 
1649.9 

1671.9 
1693.9 
1716.0 

1738.0 
1760. 1 
1782.3 


Diff. 


210 
210 
210 

211 
212 

212 

213 
213 
215 

215 
216 
216 

216 
216 
216 

217 

218 
217 

218 
218 
218 

218 
219 
219 

219 

220 
220 

220 
221 

220 

221 

222 
222 


A    (7') 


1770.64 
1781.15 
1791.72 

1802.37 
1813.10 
1823.89 

1834.76 
1845.70 
1856.71 

1867.87 
1879.08 
1890.36 

1901.70 
1913.1i 

1924.57 

1936.10 
1947.70 
1959.36 

1971.08 
1982.87 
1994.72 

2006.64 
2018.62 
2030.68 

2042.80 
2054.98 
2067.24 

2079.56 
2091.95 
2104.41 

2116.94 
2129.54 
2142.21 


Diff. 


051 
057 
065 

073 
079 
087 

094 

lOI 

116 

121 

128 
134 

141 
146 

153 

160 
166 
172 

179 

185 
192 

198  j 

206  i 
212  I 

1 

2l8| 

226  I 

232 1 


239 

246 

253 

260 
267 

274 


7(7.) 


I  0.50061 
10.50273 
I  0.50486 

!  0.50700 
0-50915 
0-51131 

0.51348 
0.51566 

0-5 


786 


0.52008 
0.52231 

0.52454 

0.52678 
0.52904 
0.53130 

0.53357 
0.53585 
0.53813 

0.54043 
0.54273 
0.54504 

0.54736 
0.54969 
0.55203 

0.55438 
0.55674 
0.55911 

0.56148 
0.56387 
0.56626 

0.56867 
0.57108 
0.57350 


Diff. 


212 
213 
214 

215 
216 
217 

218 

220 
222 

223 
223 
224 

226 
226 

227 

228 
228 
230 

230 
231 
232 

233 
234 

235 

236 

237 
237 

239  I 

239 
241 

241 

242 

244 


T(v) 


8.765 
8.791 
8.818 

8.844 
8.871 
8.897 

8.924 
8.951 
8.97.8 

9.005 
9.032 
9.060 

9.087 
9.114 
9.T42 

9.170 
9.197 
9.225  ! 

9-2531 
9.281  ! 

9-309  I 

9-337  I 
9365  i 
9-394  i 

9.422  j 

9-450 

9-479 

9-507 
9-536 
9-565 

9-593 
9.622 

9-651 


TABLE  I.— Continued. 


S{v) 


765 
764 

763 

762 
761 
760 

759 

758 

757 

756 
755 
754 

753 
752 
751 

750 
749 
748 

747 
746 

745 

744 
743 
742 

741 
740 

739 

738 
737 
736 

735 
734 
733 


1804.5 
1826.7 
1848.9 

1871.1 
1893.4 
1915-7 

1938.0 
1960.4 


2005.3 
2027.7 
2050.2 

2072.8 

2095-3 
2117.9 

2140.5 

2163. 1 
2185.8 

2208.5 

2231. 2 
2253-9 

2276.7 
2299.6 
2322.4 

2345-3 
2368.2 

2391-1 

2414.1 

2437-1 
2460.1 


Diff. 


222 
222 
222 

223 
223 
223 

224 
224 

225 

224 
225 
226 

225 
226 
226 

226 

227 

227 

227 
227 
228 


229 
228 
229 


229 
229 

230 

230 
230 
231 

2^1 


2483.2 
2506.31  231 
2529.4'  232 


A{v) 


Diff. 


2154-95 
2167.76 
2180.64 

2193-59 
2206.62 
2219.7  r 

2232.88 
2246.12 
2259.44 

2272.83 
2286.30 
2299.84 

2313-45 
2327.14 
2340.91 

2354-75 
2368.67 
2382.66 

2396.74 
2410.89 
2425.12 

2439-44 
2453-83 
2468.30 

2482.86 

2497-49 
2512.21 

2527.01 
2541.89 
2556.86 

2571.91 

2587.04 
2602.25 


281 
288 
295 

303 
309 
3^7 

324 
332 
339 

347 
354 
361 

369 

377 
384 

392 

399 

408 

415 
423 
432 

439 
447 
456 

463 
472 


/(v) 


480  0.64271 


488 

497 

505 

513 
521 

530 


0-57594 
0.57838 
0.58083 


0.58330 

0.58577 
0-58825 

0.59074 
0.59324 
0-59575 

0.59827 
0.60080 
0.60334 

0.60589 
0.60845 
0.61 103 

0.61361 
0.61620 
0.61880 

0.62142 
0,62404 
0.62667 

0.62932 
0.63198 
0.63464 

0.63732 
0.64001 


0.64542 
0164814 

0.65087 

0.65361 
0.65637 
0.65913 


Diff. 


244 
245 
247 

247 
248 

249 

250 

251 

252 

253 
254 
255 

256 

258 
258 

259 
260 
262 

262 
263 
265 

266 
266 
268 

269 

270 
271 

272 
273 
274 

276 
276 

278 


T{v) 


9.680 
9.709 
9-738 

9.767 

9-797 
9.826 

9-855 
9.885 
5.914 

9-944 

9-973 

10.003 

10.033 
10.063 
10.093 

10.123 

10.153 
10.184 

10.214 
10.244 

10.275 

10.306 
10.336 
10.367 

10.398 
10.429 
10.460 

10.491 
10.522 
10.554 

10.585 
10.616 
10.648 


Diff. 


23 


TABLE  I.— Continued. 


S(v) 


732 
731 
730 

729 

728 
727 

726 

725 
724 

723 
722 
721 

720 
719 
718 

717 
716 

714 

713. 
712 

711 
710 
709 

708 
707 
706 

705 
704 

703 

702 
701 
700 


2552.6 
2575-8 
2599.0 

2622.3 
2645.6 
2668.9 

2692.3 
2715.6 
2739.0 

2762.5 
2786.0 
2809.5 

2833.1 
2856.7 
2880.3 

2903.9 
2927,6 
2951-3 

2975-1 
2998.9 

3022.7 

3046.5 
3070.4 

3094-3 

3118.3 

3142.3 
3166.3 

3190.3 
3214.4 

3238.5 

3262.7 
3286.9 
3311.I 


Diff. 


232 
232 
233 

233 
233 
234 

233 
234 
235 

235 
235 
236 

236 
236 

236 

237 
237 
238 

238 
238 
238 

239 
239 

240 

240 
240 
240 

241 
241 
242 

242 
242 

242 


A{v) 


2617.55 
2632.94 
2648.41 

2663.97 
2679.61 
2695.34 

2711.16 

2727.07 
2743-07 

2759.16 

2775-33 
2791.60 

2807.96 
2824.41 
2840.96 

2857.60 

2874.33 
2891.15 

2908.07 
2925.08 
2942.19 

2959-39 
2976.09 

2994.09 

3011.58 
3029.17 
3046.86 

3064,66 

3082.55 
3100.54 

3118.64 
3136.84 
3155-H 


Diff, 


539 
547 
556 

564 

573 
582 


591 
600 
609 

617 

627 ; 
636  I 

645 

6551 

664 

673 
682 
692 

701 
711 
720 

730 
740 

749 

759 
769 
780 

789 

799 

810 


820 
830 
841 


/{v) 


0.66I9I 
0.66470 
0.66750 

0.67031 

0.67313 
0.67596 

0.67881 
0,68167 
0.68454 

0.68742 
0,69031 
0.69322 

0.69614 
0.69907 
0.70201 

0,70496 

0.70793 

0,71091 

0.71390 

0.7I69I 

0.71993 

0.72296 
0.72600 

0.72905 

0.73212 

0.73520 
0.73830 

0.74I4I 
0.74453 

0.74766 
0.75081 

0.75397 
0.75715 


Diff, 


279 
280 
281 

282 
283 
285 

286 
287 


289 
291 
292 

293 
294 
295 

297 
298 
299 

301 
302 
303 

304 
305 
307 

308 
310 
311 

312 
313 
315 

316 
318 
319 


T{v) 


0.679 
0.711 

0.743 

0.775 
:o.8o7 

0.839 

0.871 

0.903 
0.936 

:o.968 
1. 00 1 
1-033 

1,066 
1.099 
1-132 

1. 165 
1. 198 
1.231 

1.264 
1.297 
1-330 

1.364 
1.398 
1-432 

1.465 
1.499 
1-533 

1.567 
1.60T 
1.636 

1,670 
1,704 
1-739 


24 


TABLE  I.— Continued. 


S{v) 


699 
698 
697 

696 

695 
694 

693 
692 
691 

690 
689 
688 

687 
686 
685 

684 
683 
682 

681 
680 
679 

678 
677 
676 

675 
674 

673 
672 

67. 

670 

669 
668 
667 


3335-3 
3359-6 
3383-9 

3408.3 
3432.7 
3457-1 

3481.6 
3506.1 
3530.6 

3555-2 
3579-8 
3604.4 

3629.1 
3653-8 
3678.6 

3703-4 
3728.2 

3753-1 

3778.0 
3802.9 
3827.9 

3852.9 
3877-9 
39030 

3928.1 
3953-3 
3978.5 

4003.7 
4029.0 
4054-3 

4079.6 
4105.0 
4130.4 


Diff. 


243 
243 

244 

244 
244 

245 

245 
245 
246 

246 
246 

247 

247 
248 
248 

248 
249 
249 

249 

250 
250 

250 

25' 

251 

252 
252 
252 

253 
253 
253 

254 
254 
255 


A  (v) 


3173-55 
3192.06 
3210.67 

3229.39 
3248.22 
3267.15 

3286.19 

3305.33 
3324.58 

3343-95 
3363-42 
3383-00 

3402.70 
3422.50 
3442.42 

3462.45 
3482.60 
3502.86 

3523-24 
3543-73 
3564.34 

3585.07 
3605.91 
3626.88 

3647.96 
3669.17 
3690.50 

3711  94 

373351 

3755.21 

3777.03 
3798.98 
3821.05 


Diff. 


1 85  I 
1861 
1872 

1883 

1893 
1904 

1914 
1925 
1937 

1947 
1958 
1970 

1980 
1992 
2003 


2015 
2026 
2038 


2049  ! 
2061  I 

2073  I 

1 
2084 
2097 
2108 

2121 

2133 
2144 

2157 
2170 
2182 

2195 
2207 
2219 


I{v) 


0.76034 

0.76354 
0.76675 

0.76998 

0.77322 
0.77648 

0.77975 
0.78304 
0.78634 

0.78966 
0.79299 
0.79633 

0.79969 
0.80306 
0.80645 

0.80985 
0.81327 
0.81670 

0.82015 
0.82362 
0.82710 

0.83059 
0.83410 
0.83762 

O.84116 
0.84472 
0.84829 

0.85188 

0.85549 
O.85911 

0.86274 
0.86639 
0.87006 


Diff. 


320 
321 
3  3 

324 
326 

327 

329 
330 
332 

333 
334 
336 

337 
339 
340 

342 
343 
345 

347 
348 
349 

351 

352 
354 

356 
357 
359 

361 
362 
363 

365 
367 
369 


T{v) 


11.774 
11.809 
11.844 

11.879 
II. 914 
11.949 

11.984 

12.020 
12.055 

12.091 
12.126 
12.162 

12.198 
12.234 
12.270 

12.306 
12.342 
12.379 

12.415 

12.452 
12.489 

12.526 

12.563 
12.600 

12.637 
12.675 
12.712 

12.750 
12.787 
12.825 

12.863 
12.901 
12.939 


25 


TABLE  I.— Continued. 


Siv) 


666 
665 
664 

663 
662 
661 

660 

659 
658 

657 
656 

655 

654 

653 
652 

65' 
650 
649 

648 
647 
646 

645 
644 

643 

642 
641 
640 

639 
638 

637 

636 

635 
634 


4155-9 
4181.4 
4206.9 

4232.5 
4258.1 
4283.7 

4309-4 

4335.1 
4360.9 

4386.7 
4412.6 

4438.5 

4464.4 
4490.4 
4516.4 

4542.4 
4568.5 
4594-6 

4620.8 
4647.0 
4673.2 

4699.5 
4725-9 

4752.3 

4778.7 
4805.1 
4831.6 

4858.1 
4884.7 
4911.3 

4938.0 
4964.7 
4991.4 


Diff. 


255 
255 
256 

256 
256 

257 

257 
258 

258 

j 

259 
259 
259 

260 
260 
260 

261 
261 
262 

262 
262 
263 

264. 
264 
264 

264 
265 
265 

266 
266 
267 

267 
267 
268 


A  {7^) 


3843-24 
3865.57 
3888.02 

3910.60 

3933-31 
3956.16 

3979-13 
4002.24 
4025.48 

4048.86 
4072.37 
4096.01 

4T19.79 

4143-71 
4167.77 


Diff. 


4340.12 

4365-32 
4390.67 

4416.16 

4441.81 
4467.60 

4493-55 
4519.64 

4545-89 

4572.30 
4598.86 
4625.57 


2233 

2245 
2258 

2271 

2285 
2297 

2311 
2324 
2338 

2351 
2364 

2378 

2392 
2406 
2419 


4191.96  2434 
4216.30  2448 
4240.78  2462 

4265.40  j  2476 
4290.16  I  2491 
4315.07  2505 


2520 

2535 
2549 

2565 
2579 
2595 

2609 
2625 
2641 

2656 
2671 
2687 


/{v) 


0-87375 
0.87745 
0.88II7 

0.88490 

0.88866 
0.89243 

0.89622 
0.90002 

0.90384 

0.90768 

0.9II53 
0.9I54I 

0.91930 

0,92321 

0.92715 

0.931 10 
0.93506 
0.93904 

0.94304 

0.94706 

0.95IIO 

0.95516 
0.95923 
0.96333 

0.96745 
0.97158 
0.97574 

0.97991 

0,98410 

0.98831 
0.99254 

0.99680 
1.00107 


Diff. 


370 
372 
373 

376 
377 
379 

380 
382 
384 

385 
388 
389 

391 
394 
395 

396 
398 

400 

402 
404 
406 

407 
410 
412 

413 
416 

417 

419 
421 
423 

426 

427 
429 


T{v) 


977 
015 
053 

092 
130 
169 

208 

247 
286 

326 
365 

404 

444 
484 
524 

564 
604 

644 

684 

725 
766 

806 
847 


929 
971 

012 

053 
095 
137 


4  179 
4.221 
4-263 


Diff. 


26 


TABLE  I.— Continued. 


V 

S{zi) 

Diff. 

i 

A   (7-) 

Diff. 

I{v) 

Diff. 

T(v) 

Diff. 

633 

632 

63. 

15018.2 

i5C'45-o 
15071.9 

268 
269 
269 

4652.44 

4679.47 
4706.65 

2703 
2718  ' 

2735 

1.00536 
1.00967 
1. 01401 

431 

434 
436 

14.305 
14.348 
14.390 

43 
42 
43 

6.'?o 
629 

628 

15098.8 
15125.8 
15152-8 

270 
2701 

270  i 

4734.00 
4761.51 
4789.18 

2751 
2767 

2784 

1. 01837 
1.02274 

I.027I3 

437 
439 

442  i 

14.433 
14.476 

14.519 

43 
43 

43 

627 
626 
625 

15179-8 
15206.9 
15234.0 

271 
271 

272 

4817.02 
4845.02 
4873.18 

2800 
2816 
2833 

I  03155 
J. 03598 

1.04044 

443 
446 

448 

14.562 
14.605 
14.648 

43 
43 

44 

624 
623 
622 

15261.2 
15288.4 
15315-7 

272 
273 
273 

4901.51 
4930.00 
4958.67 

2849 
2S67 
2883 

1.04492 

1.04943 
1.05395 

451 

452 

455 

14.692 
14.735 
14.779 

43 
44 
44 

621 
620 
619 

15343.0 
15370.3 
15397.7 

273 
274 
274 

4987.50 

1  5016.51 

5045-69 

2901 
2918 

2935 

1 1.05850 
1.06307 
1.06766 

457 
459 
461 

14.823 
14.867 
14.91 1 

44 
44 
45 

618 
617 
616 

15425. 1 
15452.6 
15480.1 

275 

275 
276 

5075-04 
5104.57 
5134-27 

2953 
2970 
2988 

1.07227 
1.07690 
1. 08156 

463 
466 
468 

,14.956 

15.000 

15.045 

44 
45 

45 

615 
614 

613 

15507.7 
15535.3 
15563.0 

276 

277 
277 

5164.15 
5194.21 
5224.44 

3006 
3023 
3042 

1.08624 
1.09095 
1.09568 

471 
473 
475 

15.090 

15-135 
15.180 

45 
45 
45 

612 
611 
610 

T5590.7 
15618.4 
15646.2 

277 

1  278 

278 

5254-86 

i  5285.46 

5316.24 

3060 
3078 
3097 

1 10043 

1. 10520 

I.IIOOO 

477 
48c 
482 

15-225 
15.270 
T5-316 

45 
46 
45 

609 
608 
607 

1  15674.0 
15701.9 
15729.8 

279 
279 

280 

5347-21 
5378.36 
5409-71 

31^5 
3135 
3153 

1  I.II482 
I.II966 
1. 12452 

484 
486 

489 

15-361 

15-407 
15.453 

46 
46 
46 

606 
605 
604 

15757-8 
15785.8 
15813-9 

280 
i  281 
1  281 

1 

!  5441.24 

5472.95 
5504.86 

3171 

3191 
3210 

I.I294I 

1. 13433 

1 1.13927 

492 
494 
497 

■ 
15-499 
15-546 
15.592 

47 
46 
46 

603 
602 
601 

15842.0 
15870. 1 
15898.3 

281 
282 

'  283 

1553696 
i  556926 
'5601.75 

3230 

3249 
3268 

1. 14424 
1. 14923 
1-15425 

499 

502 

504 

15.638 
15.685 
15.732 

47 
47 
47 

27 


TABLE  I.— Continued. 


V 

S{v) 

Diff.  I 

1 
1 

A  (v)        Diff. 

I{v) 

Diff. 

T{v) 

Diff. 

600 

599 
598 

15926.6 

15954.9 
15983-2 

283 
283 
284 

5634.43  I  3288 
5667.31  J3309 
5700.40  3329 

1. 15929 
1. 16435 
1.16944 

506 
509 

5'2 

15.779 
15.826 

15.873 

47 
47 
48 

597 
596 
595 

160IT.6 
1 6040. 1 
16068.6 

285 
285 

285  1 

5733-69 

5767.18 
5800.87 

3349 
3369 
3389 

1. 17456 
1. 17970 

1. 18487 

5H 
517 
519 

15.921 
15.968 
16.016 

47 
48 
48 

594 
593 
592 

16097. 1 
16125.7 
16154-4 

286 

287 
287 

5834-76 
5868.85 
5903.16 

3409 
3431 
3451 

1. 19006 
1. 19528 
1.20053 

522 
525 

527 

16.064 
16. 113 
16. 16  I 

49 

48 
48 

591 
590 
589 

16183.I 
16211.8 
16240.6 

287  1 

288 

288 

5937-67 

5972.39 
6007.32 

3472 
3493 
3515 

T. 20580 
I.2IIIO 
1. 21643 

530 
533; 

535  , 

16.209 
16.258 
16.307 

49 
49 
49 

588 

587 
586 

16269.4 
16298.3 
16327.2 

289 
289 
290 

6042.47 

6077.83 
6113.41 

3536 
3558 
3579 

I. 22178 
I. 22716 
1.23257 

538 
541 
544 

16.356 
16.405 

16.454 

49 
49 
50 

585 
584 
583 

16356.2 
16385.2 
16414.3 

290 
291 
291 

6149.20 
6185.22 
6221.46 

3602 
3624 
3646 

I. 23801 
1.24348 
1.24897 

547 
549 
552 

16.504 

16.553 
16.603 

49 

50 
50 

582 
581 
580 

16443.4 
16472.6 
16501.8 

292 
292 
293 

6257.92 
6294.61 
6331-52 

3669 
3691 
3714 

1.25449 
1.26004 
1.26562 

555 
558 
561 

16.653 
16.704 
16.754 

51 

50 
51 

579 

578 

577 

16531.1 
16560.4 
16589.8 

293 
294 
294 

6368.66 
6406.01 
6443-63 

3735 
3762 

3783 

1. 27123 

1.27687 
1.28253 

564 

566 

570 

16.805 
16.855 
16.906 

50 
51 

52 

576 

575 
574 

16619.2 
16648.7 
16678.2 

295 
295 
296 

6481.46 

65^9.52 

6557.82 

3806 
3830 
3854 

1.28823 
1.29396 
I. 29971 

573 
575 
579 

16.958 
17.009 
17.060 

51 
51 

52 

573 
572 
571 

16707.8 
16737.4 
16767. 1 

296 

297 
298 

6596.36 

6635.14 
6674.16 

3878 
3902 
3926 

1.30550 
I.3II3' 
I.31716 

581 

585 
588 

17.112 

17.164 

i  17.216 

52 
52 
52 

570 

569 
568 

16796.9 
16826.7 
16856.6 

298 

299 
299 

671342 

6752.93 
6792.68 

3951 
3975 
4000 

1.32304 
1-32895 
1-33489 

591 
594 
597 

17.268 
17.320 
17-373 

52 
52 

28 


TABLE  I.— Continued. 


S{v) 


567 
566 

565 

564 
563 
562 

561 
560 

559 

558 
557 
556 

555 
554 
553 

552 

551 
550 

549 
548 
547 

546 

545 
544 

543 
542 
541 

540 
539 
538 

537 
536 
535 


6886.5 
6916.4 
6946.4 

6976.5 
7006.6 
7036.8 

7067.0 
7097.3 
7127.6 

7158.0 
7188.4 
7218.9 

7249.4 
7280.0 
7310.7 

7341-4 

7372.2 
7403.0 

7433-9 
7464.8 

7495-8 

7526.8 
7557.9 
7589-1 

7620.3 
7651.6 
7682.9 

77H-3 
7745-8 
7777-3 

7808.9 

7840-5 
7872.2 


Diff. 


299 
300 
301 

301 

302 
302 

303 
303 
304 

304 

305 
305 

306 
307 
307 

308 
308 
309 

309 
310 
310 

311 
312 
312 

313 
313 
314 

315 
315 
316 

316 
317 
317 


Aiv) 


6832.68 
6872.93 
6913-43 

6954.18 
6995.19 
7036.46 

7077.99 
7119.78 
7161.83 

7204.15 
7246.73 
7289.58 

7332.71 
7376.11 
7419.78 

7463.74 
7507-97 
7552.48 

7597.28 
7642.36 
7687.73 

7733-39 
7779-34 
7825.58 

7872.12 
7918.96 
7966.12 

8013-55 
8061.30 
8109.36 

8T57-73 
8206.41 
8255.41 


Diff. 


4025 
4050 
4075 

4101 
4127 
4153 

4179 
4205 
4232 

4258 
4285 

43^3 

4340 
4367 
4396 

4423 
4451 
4480 

4508 

4537 
4566 

4595 
4624 

4654 

4684 
4716 
4743 

4775 
4806 

4837 

4868 
4900 
4932 


/{v) 


34086 
34686 
35290 

35897 
36507 
37120 

37736 
38356 
38979 

39606 

40236 

40869 

41506 

42146 

42789 

43436 

44087 
44741 

45399 
46060 

46725 

47394 
48066 

48742 

49422 
50106 
50793 

51484 
52179 
52878 

53581 
54287 
54998 


Diff. 


600 
604 
607 

610 
613 
616 

620 
623 
627 

630 

637 

640 
643 
647 

651 
654 

658 

661 
665 
669 

672 
676 
680 

684 
687 
691 

695 
699 

703 

706 
711 
715 


Tiv) 


7.425 
7-478 
7-531 

7-584 
7.638 
7.691 

7.745 
7-799 
7.853 

7-908 
7.962 
8.017 

8.072 
8.127 
8.183 

8.238 
8.294 
8.350 

8.406 
8.462 
8.519 

8.576 

8.633 
8.690 

8.747 
8.805 

8.921 

8.979 
9.038 

9.096 
9-155 
9-215 


Diff. 


29 


TABLE  I.— Continued. 


S{v) 


534 
533 

532 

531 

530 
529 

528 
527 
526 

525 
524 
523 

522 

521. 
520 

519 
518 

517 

516 

5^5 
514 

513 
512 

511 

510 
509 
508 

507 
506 

505 

504 
503 
502 


7903.9 

7935-7 
7967.6 

7999-5 
8031.5 
8063.5 

8095.6 
8127.8 
8160.0 

8192.3 

8224.7 
8257.1 

8289.6 
8322.1 
8354-7 

8387-4 
8420.1 
8452.9 

8485.7 
8518.6 

8551-6 

8584-7 
8617.8 
8651.0 

8684.2 
8717-5 
8750-9 

8784.3 
8817.8 
8851.4 

8885.0 
8918.7 
8952.5 


Diff 


318 
319 
319 

320 
320 
321 

322 
322 
323 

324 
324 
325 

325 
326 

327 

327 
328 
328 

329 
330 

33^ 

33'^ 
332 
332 

333 
334 
334 

335 
33^ 
33^ 

337 
33^ 
338 


A'iv) 


8304.73 
8354.36 
8404.32 

8454.61 
8505.22 
8556.16 

8607.44 
8659.06 
8711.01 

8763.30 
8815.94 
8868.92 

8922.25 

8975.93 
9029.97 

9084.36 
9139.11 
9194.23 

9249.71 

9305.56 
9361.79 

9418.39 
9475-38 
9532.74 

9590.49 
9648.62 
9707.15 

9766.06 
9825.38 
9885.09 

9945.21 
10005.74 
10066.67 


Diff. 


4963 
4996 
5029 

5061 

5094 
5128 

5162 

5195 
5229 

5264 
5298 
5333 

5368 
5404 
5439 

5475 
5512 

5548 

5585 
5623 
5660 

5699 
5736 

5775 

5813 
5853 
5891 

5932 

5971 
6012 

6053 
6093 

6134 


I{v) 


55713 
56431 
57154 

57881 

58612 

59347 

60086 
60830 
61578 

62330 
63086 
63847 

64612 
65381 
66155 

66933 
67716 
68504 

69296 
70092 
70894 

71700 
72510 
73326 

74146 

74971 
75801 

76636 
77476 
78321 

79171 
80026 
80886 


Diff. 


718 

723 

727 

731 
735 
739 

744 
748 
752 

756 
761 

765 

769 

774 
778 

783 
788 
792 

796 
802 
806 

810 
816 
820 

825 
830 
835 

840 

845 
850 

855 
860 

865 


T{v) 


19.274 
J9.334 
19-394 

^9-454 
19.514 
19-574 

19-635 
19.696 

19-757 

19.819 
19.881 
19.943 

20.005 
20.067 
20.130 

20.193 
20.256 
20.319 

20.383 
20.447 
20.511 

20.575 
20.640 
20.705 

20.770 
20.835 
20.901 

20,967 
21.033 
21.099 

21.166 
21.233 

21,300 


30 


TABLE  I.— Continued. 


501 
500 

499 

498 

497 
496 

495 
494 
493 

492 
491 
490 

489 
488 
487 

486 
485 
484 

483 
482 
481 

480 

479 
478 

477 
476 

475 

474 

473 
472 

471 

470 
469 


■S(v) 


3 

9020.2 
9054.2 

9088.2 
9122.3 
9156.4 

9190.6 
9224.9 
9259-3 

9293.8 

9328.3 
9362.9 

9397.6 

9432.3 
9467.1 

9502.0 

9536.9 
9572.0 

9607.1 
9642.2 
9677-5 

9712.8 
9748.2 
9783.6 

9819. 1 

9854.7 
9890.4 

9926.2 
9962.0 
9997-9 

20033.9 
20070.0 
20106,2 


Difif. 


339 

340 
340 

341 
341 
342 

343 
344 
345 

345 
346 
347 

347 
348 
349 

349 
351 
351 

351 
353 
353 

354 

354 
355 

356 

357 
358 

358 

359 
360 

361 
362 
362 


A{v) 


0128.01 
0189.78 
0251.9 

0314.5 
0377.6 
0441.0 

0504.9 

0569-3 
0634.1 

0699.3 
0765.0 
0831. 1 

0897.6 
0964.7 
1032.2 

1 100. 1 
1168.6 
1237-5 

1307.0 
1376.9 
1447-2 

1518.1 
1589.4 
1661.3 

1733-7 
1806.6 
1880.0 


1953-9 

2028.4 
2103.4 

2178.9 
2254.9 
2331-5 


Diff. 


6177 

6219 

626 

631 
634 
639 

644 

648 

652 

657 
661 
665 

671 

675 
679 

685 
689 

695 

699 

703 
709 

•713 
719 

724 

729 
734 
739 

745 
750 
755 

760 
766 
771 


/{v) 


.81751 

.82622 

.83498 
•84379 

-85265 
-86157 

•87054 
-87957 
.88865 

.89778 

.90697 

.91622 

•92552 
-93488 
.94430 

•95378 
.96332 
.97292 

.98258 

.99230 

2.00207 


2.0II90 
2.02180 
2.03176 


2.04179 
2.05188 

2,06203 

2.07225 

2.08253 

2.09288 
2.10329 

2.II376 

2.12430 


Diff, 


871 
876 


886 
892 
897 

903 
908 

913 

919 

925 
930 

936 
942 
948 

954 
960 
966 

972 
977 
983 

990 

996 
T003 

1009 
1015 

1022 

1028 

1035 
1041 

1047 

1054 
1061 


T{v) 


21.367 

21.435 
21.503 

21.572 
21.641 
21.710 

21.779 
21.848 
2T.918 

21.988 
22.058 
22.128 

22.199 
22.270 
22.341 

22.413 
22.485 
22.557 

22,630 
22,703 
22.776 

22.849 
22.923 
22.997 

23.071 
23.146 
23.221 

23.296 
23.372 
23.448 

23-524 
23.601 
23.678 


Diff. 


31 


TABLE  I.— Continued, 


V 

S{v) 

Diff. 

A{v) 

Diff. 

/{v) 

Diff. 

T{v) 

Diff. 

468 
467 
466 

20142.4 
20J78.7 
20215.0 

363 
365 

12408.6 

'12486.3 

12564.6 

777 
783 
788 

2.13491 
2.14559 
2.15635 

ic68  i 
1076  i 
1082 

23.755 
23.833 
23.911 

78 
78 
78 

465 
464 

463 

20251.5 
20288.0 
20324.7 

365 
367 
367 

12643.4 
12722.8 
12802.7 

794 
799 
805 

2.16717 
2.17806 
2.18902 

1089  1 
1096 
1 104 

23.989 
24.068 
24.147 

79 
79 
79 

462 
461 
460 

20361.4 
20398.1 
20435.0 

367 
369 
369 

12883.2 
12964.3 
13045-9 

811 
816 
822 

2.20006 
2.21116 

2.22233 

i 
1110 

1117 

1124 

24.226 
24.306 
24.386 

80 
80 
80 

459 
458 
457 

20471.9 
20508.9 
20546.0 

370 
371 
371 

13128. 1 
13211.O 
13294.4 

829 

834 
841 

2.23357 

2.24489 
2.25629 

1132 
1140 
1147 

24.466 

24.547 
-  24.628 

81 
81 

82 

456 
455 
454 

20583.1 
20620.4 
20657.7 

373 
373 

374 

13378.5 

134633 
13548.6 

848 
853 
859 

2.26776 

2.27931 

2.29094 

1155 
1163 
1171 

24.710 
24.792 
24.874 

82 
82 
82 

453 
452 
451 

20695.1 
20732.6 
20770.2 

375 
376 
377 

13634.5 
13721.1 
13808.3 

866 

872 
878 

2.30265 

2.31443 
2.32628 

1178 
1185 
1193 

24.956 

25-039 
25.122 

^3 
83 
84 

450 
449 
448 

20807.9 
20845.6 
20883.4 

377 
378 
380 

13896.1 
13984.6 
14073.7 

885 
891 
898 

2.33821 
2.35022 
2.36232 

1201 
1210 
1218 

25.206 

1  25.290 

25.374 

84 
84 
85 

447 
446 

445 

20921.4 
20959.4 
20997.4 

380 
380 
382 

'4163.5 

14254.0 

14345. i 

905 
911 
919 

2.37450 

2.38676 
2.39911 

1226 
1235 
1243 

1  25.459 
25.544 
25.629 

85 
85 
86 

444 

443 
442 

21035.6 
21073.9 
21112.2 

385 

14437.0 

14529.5 
14622.7 

925 
932 
939 

2.41154 
2.42405 
2.43665 

1251 
1260 
1268 

i  25.715 
25.801 
25.888 

86 
87 
87 

441 
440 
439 

21150.7 
21189.2 
21227.8 

385 
386 

387 

14716.6 
14811.2 
14906.5 

946 

953 
960 

2.44933 

2.46209 

2.47494 

1276 
1285 
1294 

25.975 
26.062 
26.150 

87 
88 
88 

438 
437 
436 

21266.5 

213^5-3 
2i34'4-2 

388 
389 
389 

15002.5 

15099.3 
15196.8 

968 

975 
982' 

2.48788 
2.50091 

2.51404 

1303 
1313 
1322 

26.238 

26.327 

'  26.416 

89 
89 
89 

32 


TABLE  I.— Continued. 


S{v) 


21383. 1 
21422.2 
21461.4 

21500.6 
21540.0 
21579-4 

21618.9 
21658.5 
21698.2 

21738.0 
21777.9 
21817.8 

21857.9 
21898.1 
21938.4 

21978.7 
22019.1 
22059.6 


Diff. 


22100.2 
22140,9 
22 


81.8 


22222.7 
22263.7 
22304.8 

22346.1 

^2387.4 
22428.8 

22470.4 
225  12.0 

22553-7 

22595.6 
22637.5 
22679.6 


391 
392 
392 

394 
394 
395 

396 
397 
398 

399 
399 

401 

402 
403 
403 

404 

405 
406 

407 
409 
409 

410 
411 
413 

413 
414 
416 

416 
417 
419 

419 
421 

422 


A{v) 


5295-0 
5394.0 
5493-7 

5594-2 
5695-4 
5797-3 

5900.0 
6003.5 
6107.9 

6213. 1 
6319.1 
6425.9 

6533-5 
6641.9 
6751.2 

6972.2 
7084.1 

7196.8 
7310.5 
7425.0 


7540.5 
7656.8 

7774.1 

7892.2 
8011.3 
813^-3 

8252.4 

8374.4 
8497.4 

8621.4 
8746.4 
8872.3 


Diff. 


990 

997 

1005 

012 
019 

027 

035 
044 
052 

060 
068 
076 

084 
093 


T09 
119 
127 

137 
145 
155 

163 

173 
181 

191 
200 
211 

220 
230 
240 

250 

259 
270 


I{v) 


2.52726 

2.54057 
2.55397 

2.56746 
2.58104 
2.59471 

2.60848 
2.62235 

2.63632 

2.65039 
2.66456 

2.67883 

2.69320 
2.70767 

2.72225 

2.73692 
2.75169 

2.76658 
2.78158 

2.79668 
2,81190 

2,82723 
2.84267 

2.85822 

2.87388 
2.88965 
2.90554 

2.92155 
2.93768 

2.95393 
2.97030 

2.98679 

3.00341 


Diff. 


33^ 

340 
349  i 

358 
367 

377 

387 
397 
407 

417 

427 

437 

447 

458 
467 

477 
489 
500 

510 

522 

533 

544 

555 
566 

577 
589 
601 

613 
625 

637 

649 
662 

674 


T{v) 


26.505 

26.595 
26.685 

26.776 
26.867 
26.959 

27.051 

27.143 
27.236 

27.329 
27.423 
27.517 

27.612 

27.707 
27.803 

27-899 
27-995 
28.092 

28.189 
28.287 
28.385 

28.484 
28.583 
28.683 

28.783 
28.884 
28.985 

29.087 
29.189 
29.292 

29-395- 
29.499 
29,603 


33 


TABLE  I.— Continued. 


V 

Siv) 

Difif. 

A{v) 

Diff. 

I{v) 

Diff. 

Tiy) 

Diff. 

402 
401 
400 

22721.8 
22764.0 
22806.4 

422 
424 
424 

18999.3 
19127.3 
19256.2 

1280 
1289 

1300 

3.02015 
I  3-03701 
'  3-05399 

1686 
1698 
1710 

29.708 
29.813 
29.919 

105 
106 
106 

34 


TABLE  II. 

For  Spherical  Projectiles. 


V 

S{v) 

Diff. 

A(v) 

Diff.  1 

I{v) 

Diff. 

T{v) 

Diff. 

2000 

0 

25 

0.00 

I  i 

0.00000 

40 

0.000 

12 

1990 
1980 

25 
49 

24 

25 

O.OI 

0.02 

I ! 
2 

00040 
00080 

40 
41 

0.012 
0.025 

13 

12 

1970 
i960 
1950 

74 

99 
124 

25 
25 
26 

0.04 
0.08 
0.13 

4 

5 

51 

0.00121 
00163 
00205 

42 
42 
43 

0.037 
0.050 
0.063 

13 
13 
13 

1940 
1930 
1920 

150 
175 
201 

25 
26 

25 

0.18 
0.25 
0.33 

7J 
8  1 

9 

0.00248 
00292 
00336 

44 
44 
45 

0.076 
0.089 
0.102 

13 

13 
14 

I9IO 
1900 
1890 

226 

252 
278 

26 
26 
26 

0.42 

0.53 
0.65 

13 

0.00381 
00427 
00473 

46 
46 

47 

0.1 16 
0.129 
0.143 

13 
14 
14 

1880 
1870 
i860 

304 

357 

26 

27 
26 

0.78 
0.92 
1.07 

14 
15 
17 

0.00520 
00568 
00617 

48 
49 
49 

0.157 
0.171 
0.185 

14 
14 
14 

1850 
1840 
1830 

383 
409 

436 

26 

27 
27 

1.24 

1.43 
1.63 

19 

20 

21 

0.00666 
00716 
00767 

50 
51 

52 

0.199 
0.214 
0.228 

15 
14 
15 

1820 
181O 
1800 

463 
490 

517 

27 

27 
28 

1.84 
2.07 
2.31 

23 
24 
26 

0.00819 
00872 
00926 

53 

0.243 
0.258 
0.273 

15 
15 
15 

1790 

1780 
1770 

545 
572 
600 

27 
28 
28 

2.57 
2.84 

3.14 

27 
30 
31 

0.00981 
01036 
01093 

55 
57 
57 

0.288 
0.304 
0.319 

16 

15 
16 

1760 

1750 
1740 

628 
656 
684 

28 
28 
28 

3.45 
3-78 
4-13 

35 
37 

0.01150 
01209 
01268 

59 
59 
61 

0.335 
0.351 
0.367 

16 
16 
16 

1730 
1720 
I7IO 

712 
741 
769 

29 
28 

29 

4.50 
4.89 
5-30 

39 
41 
43 

0.01329 
01390 
01453 

61 

(>z 

64 

0.383 
0.400 
0.416 

17 
16 

17 

35 


TABLE  IL— Continued. 


V 

S{v) 

Diff. 

A{v) 

Diff; 

I{v) 

Diff. 

T{v) 

Diff. 

1700 
1690 
1680 

798 
827 
856 

29 
29 
30 

5-73 
6.18 
6.65 

45 
47 
50 

O.OI5I7 
01582 
01648 

65 

66 
67 

0.433 
0.450 
0.468 

17 

18 

17 

1670 
1660 
1650 

886 
915 
945 

1 
29 

30 
30 

7.15 
7.67 
8.21 

52 
54 
56 

O.OI7I5 

01783 
01853 

68 
70 
71 

0.485 

0-503 
0.521 

18 
18 
t8 

1640 
1630 
1620 

975 
1005 
1036 

30 
3T 
30 

8.77 
9-35 
9-97 

58 
62 
64 

0.01924 
01996 
02070 

72 
74 
75 

0.539 
0.558 
0.576 

19 
18 

19 

I6I0 
1600 

1590 

1066 
1096 
T127 

30 
31 
31 

10.61 
11.27 
11.96 

66 
69 

72 

0.02145 
02222 
02300 

79 

0.595 
0.614 
0.633 

19 
19 

20 

1580 
1570 

1560 

1158 
1189 

1220 

31 
31 

32 

12.68 

1344 
14.22 

76 

78 
82 

0.02379 
02460 

02542 

81 

82 
84 

0.653 
0.673 
0.693 

20 
20 
20 

1550 

1540 
1530 

1252 
1284 
1316 

32 
32 
32 

15-04 
15.90 
16.78 

86 
88 
92 

0.02626 
02712 

02799 

86 
87 
89 

0.713 
0.734 
0.755 

21 
21 
21 

1520 
I5I0 
1500 

1348 
1380 

1413 

32 

17.70 
18.65 
19.63 

95 
98 

100 

0.02888 
02979 

03072 

91 
93 
94 

0.776 
0.797 
0.819 

21 
22 
22 

1490 
1480 

1470 

1446 
1479 
1512 

Z2> 
7>Z 
34 

20.63 
21.68 

22.77 

105 
109 
114 

0  03166 
03262 
03360 

-  96 
98 

ICI 

0.841 
0.863 
0.885 

22 
22 
23 

1460 

1450 
1440 

1546 
1580 
1614 

34 
34 
34 

23.91 

i  25.10 
26.34 

! 

119 
124 

128 

0  03461 

03564 
03669 

103 
105 
107 

0.908 
0931 
0.955 

23 
24 
24 

1430 

T420 
I4I0 

1648 
1682 
1717 

34 

35 
35 

27.62 
1    28.95 

133 
138 
143 

0.03776 

03885 
03997 

109 
112 
114 

0.979 
T.003 
1.028 

24 

25 

25 

1400 

1390 
1380 

1752 
1787 
1823 

35 
35 

31.76 

33-25 
34-79 

149 

154 
160 

0.041 1 1 
04227 
04346 

116 
119 
122 

1.053 
1.079 
1. 105 

26 
.6 
26 

36 


TABLE  II.— Continued. 


V 

S{v) 

Diff. 

A{v) 

Diff. 

I{v) 

Diff. 

T{v) 

Diff. 

1370 
1360 

1350 

1858 
1894 
1931 

36 
37 
36 

36.39 
38.03 
39-73 

164 

170 

175 

0.04468 

04592 
04719 

124 
127 
129 

1. 131 

1. 158 
1. 185 

27 
27 
27 

1340 

1330 
1320 

1967 
2004 
2041 

37 

37 
37 

41.48 
43-29 
45-14 

181 

185 
191 

0.04848 
04981 

05II7 

136 
139 

1. 212 
1.239 
1.267 

27 
28 
27 

I3I0 
1300 
1290 

j    2078 

2Tl6 
2T54 

38 
38 

47-05 
49.01 

51-04 

196 

203 
212 

0.05256 

05398 
05542 

142 
144 

148 

1.294 
1.322 
I-351 

28 

29 
30 

1280 
1270 
1260 

2192 
2231 
2269 

39 
38 
39 

53-16 

55-37 
57.67 

221 
230 
240 

0.05690 

05842 
05998 

152 
156 
160 

1.38T 
1.411 
1.442 

30 
31 
31 

1250 

1240 
1230 

2308 

2348 

1     2388 

40 
40 
40 

60.07 
62.56 
65.14 

249 
258 
267 

0.06158 

06323 

06492 

165 

169 
174 

1.473 

1-505 
1.538 

32 
33 

1220 

I2IO 

1200 

2428 
2470 
2512 

42 
42 
22 

67.81 
70.59 
73-54 

278 

295 
156 

0.06666 

06846 
07033 

180 

187 

97 

1.571 
1.605 
1.640 

34 

35 
18 

I  I  90 

2534 
2556 
2578 

22 
22 
22 

75-IO 
76.70 

78.32 

t6o 
162 
165 

0.07130 
07229 

07329 

99 
100 

102 

1.658 
1.676 
1.694 

18 
18 
18 

I  180 

II75 
1170 

1     2600 
2623 
2646 

'23 
23 
23 

79-97 
81.66 

83-39 

169 

173 
177 

0.07431 
07535 

07641 

104 
106 
108 

1.712 
I-731 
I-751 

19 

20 

19 

I165 
I  160 

i'55 

j 

i     2669 
2692 
2715 

23 
23 
24 

85.16 
86.98 
88.84 

182 
186 
190 

0.07749 

07859 
07972 

no 
113 
115 

1.770 
1.790 
1. 810 

20 
20 
21 

1150 

1 145 

1 140 

2739 
2763 
2787 

24 

24 

25 

90.74 
92.69 
94.68 

195 
199 

205 

0.08087 
08204 

08324 

117 
120 
122 

1.831 
1.852 
1-873 

21 
21 

22 

1130 
1125 

2812 
2837 
2861 

25 
24 

25 

96.73 

98.82 

100.97 

209 

215 
221 

0.08446 
08570 
08697 

124 
127 
130 

1.895 

1-917 
1.940 

22 
23 
23 

37 


TABLE  II.— Continued. 


V 

Si^v) 

Diff. 

A{v) 

Diflf. 

I{v) 

Diff. 

T{v) 

Diff. 

II20 
IITO 

2886 
2912 
2938 

26 
26 
26 

103.18 

J05-44 

107.77 

226 
233 
239 

0.08827 
08959 

09094 

132 

'35 
138 

1.963 
1.986 
2.009 

23 

24 

IIO5 
IIOO 

1095 

2964 
2991 
3017 

27 
26 
27 

110.16 
1 12,62 
115-13 

246 

251 
259 

0.09232 

09373 
09516 

141 
'43 
147 

2.033 

2.057 
2.081 

24 

24 
25 

1090 
1085 
1080 

3044 
3071 

3099 

27 
28 
28 

117.72 
120.38 
123.13 

266 

■275 
283 

0.09663 
09812 

09965 

149 

153 
156 

2.106 

2.132 
2.158 

26 
26 
26 

1075 

1070 
1065 

3127 

3155 
3184 

28 
29 
29 

125.96 
128.87 
131.87 

291 
300 
308 

0.10121 
10280 

10443 

159 
163 
166 

2.184 
2.210 

2.237 

26 

27 
28 

io6o 

1055 
1050 

3213 
3243 
3273 

30 
30 
30 

134.95 
138.12 
141.38 

317 
326 

338 

0.10609 

10-79 
10952 

170 

173 
177 

2.265 
2.293 
2.321 

28 
28 
29 

1045 
T040 

1035 

3364 

30 
31 
31 

144.76 
148.22 
151-77 

346 
355 
364 

0.11129 
11310 
1 1495 

181 

185 
189 

2.350 

2.379 
2.409 

29 
30 
31 

1030 
1025 
1020 

3395 
3427 
3459 

32 
32 
32 

155-41 

159-15 
162.99 

374 
384 
394 

0.11684 

II877 
12074 

193 
197 

202 

2.440 
2.471 
2.502 

31 
31 
32 

1015 

lOIO 

1005 

3491 

3524 

3557 

Z2> 
34 

166.93 
170.99 
175-17 

406 
418 
430 

0.12276 
12482 

12693 

206 
211 
215 

2.534 
2.566 

2.599 

32 

1000 

995 
990 

3591 
3625 
3660 

34 

35 
35 

179-47 
183.90 
188.46 

443 
456 

470 

0.12908 

13128 
13354 

220 
226 
231 

2.632 
2.665 
2.699 

zz 

34 
ZS 

985 
980 

975 

3695 
3731 
3767 

36 

193.16 
198.00 
202.98 

484 
498 
513 

0.13585 
13821 

14062 

236 
241 
246 

2.734 

2.770 
2.806 

36 
37 

970 

965 
960 

3803 
3840 

3877 

37 
37 

38 

208.1 1 
213.40 
218.86 

529 
546 
563 

0.14308 
14560 
14818 

252 

258 
264 

2.843 
2.881 
2.920 

38 
39 
39 

38 


TABLE  II.— Continued. 


V 

S{v) 

Diff. 

A{v) 

Diff. 

I{v) 

Diff. 

T{v) 

Diff. 

955 
950 
945 

39^5 
3953 
3992 

38 
39 
39 

224.49 
230.29 
236.29 

580 
600 
620 

0.15082 

15352 
15628 

270 
276 
283 

2-959 
2.999 

3.040 

40 

41 

42 

940 

935 
930 

4031 
4070 
41 10 

39 

40 

41 

242.49 
248.86 
255-43 

637 
657 
676 

O.I59II 

I620I 
16498 

290 
297 
304 

3.082 
3-125 
3.168 

43 
43 
44 

925 
920 

915 

4151 
4192 

4234 

41 

42 

43 

262.19 
269.17 
276.37 

698 
720 
743 

0.16802 

17113 

17432 

311 
319 
327 

3-212 
3-257 
3-303 

45 
46 

47 

910 

905 
900 

4277 
4320 

4363 

43 
43 
44 

283.80 
291.47 
299.40 

767 

793 
819 

0.17759 

18094 

18437 

335 
343 

352 

3-350 

3-397 

3.445 

47 
48 

49 

895 
890 
885 

4407 

4451 
4496 

44 
45 
46 

307-59 
316.04 

324-77 

845 
873 
901 

0.18789 
I9I49 
I95I8 

360 

369 

378 

3-494 
3.544 
3.595 

50 
51 
52 

880 

875 
870 

4542 

4589 
4636 

47 
47 
48 

333-78 
343-06 
352.67 

928 
961 
997 

0.19896 
20283 
20680 

387 
397 

407 

3-647 

3.700 

3-754 

53 

54 
55 

865 
860 
855 

4684 
4732 
4781 

48 
49 
49 

362.64 
372.96 

1032 
1064 
1099 

0.21087 

21505 
21933 

418 
428 
439 

3-809 
3.865 
3.922 

56 

57 
58 

850 

845 
840 

4830 
4880 
4931 

50 
5» 

52 

394.59 
405-96 
417.71 

"37 

1175 
1216 

0.22372 

22823 

23285 

451 
462 

476 

3-980 

4.039 
4.100 

59 
61 
61 

835 
830 

825 

4983 
5036 
5089 

53 

53 
54 

429.87 
442.45 
455-47 

1258 
1302 
1347 

0.23761 
24248 
24746 

487 
498 

511 

4. 161 

4.224 
4.288 

64 

820 

815 
810 

5143 
5198 
5253 

55 
55 
56 

468.94 
482.89 
497-33 

1395 
1444 

1495 

0.25257 
25783 

26323 

526 

540 

553 

4-354 
4-421 
4.489 

67 
68 

70 

805 
800 
795 

5309 
5366 
5424 

57 
58 
59 

512.28 

527-77 
543-81 

1549 
1604 
1661 

0.26876 
27444 

28031 

568 

587 
601 

4-559 
4.630 
4.702 

71 

72 
74 

39 


TABLE  II.— Continued. 


V 

Siv) 

Diff. 

A{v) 

Diff. 

nv) 

Diff. 

T{v) 

Diff. 

79Q 

785 
780 

5483 
5542 
5602 

59 
60 
61 

560.42 

577-64 
595-48 

1722 
1784 
1849 

0.28632 
29249 
29883 

617, 

634 

650; 

4.776 
4.852 
4-929 

76 

77 
79 

775 
770 

765 

5663 

5725 
5788 

62 

63 
64 

613-97 
653.01 

1916 
1988 
2062 

0.30533 
31203 
31891 

670 
688 

707 

5.008 
5.088 
5-170 

80 
82 
84 

760 

755 
750 

5852 
5917 
5983 

65 
66 

67 

673-63 
695.01 
717  19 

2138 
2218 

2303 

0.32-598 
33325 
34073 

727 
748 
770 

5254 
5-340 
5-427 

86 

87 
90 

745 
740 

735 

6050 
6118 
6187 

68 
69 
69 

740.22 
764.11 
788.91 

2389 
248c 

2574 

0.34843 
35634 
36448 

791 
814 
837 

5-517 
5-608 

S-701 

91 
93 
96 

730 

725 
720 

6256 
6327 
6399 

71 

72 
73 

814.65 
841.38 
869.14 

2673 
2776 
2882 

0.37285 
38146 
39033 

861 
887 
912 

5-797 
5.894 
5-994 

97 
100 

102 

715 
710 

705 

6472 
6546 
6621 

74 

75 
77 

897.96 
927.92 
959.07 

2996 

3115 

3238 

0.39945 

40885 

41853 

940 
968 

995 

6.096 
6.200 
6.306 

104 
106 
109 

700 

695 
690 

6698 
6776 
685s 

78 

79 

80 

991.45 
1025.2 
1060.2 

3366 
350 
364 

0.42848 

43872 
44926 

1024 

1054 
1089 

6.415 
6.526 
6.640 

III 
ri4 
116 

685 
680 

675 

6935 
7016 
7098 

81 

.  82 

84 

1196.6 

1134.4 
1173.8 

378 
394 
409 

0.46015 

47143 
48302 

1128 

1159 
1192 

6.756 
6.875 
6.997 

1 

119 
122 
125 

670 
665 
660 

7182 
7267 
7354 

85 
87 
88 

1214.7 

1257.4 
1301.8 

427 
444 
463 

0.49494 
50722 

51989 

1228 
1267 
1307 

7.122 

7-249 
7-380 

127 

131 
134 

655 
650 

645 

7442 

7531 
7622 

89 

91 
92 

1348.1 

1396.3 
1446.5 

482 
502 
523 

0.53296 
54645 
56037 

1349 
1392 

1436 

7-5M 

7-651 

1   7.79T 

137 
140 

143 

640 

635 
630 

7714 
7808 
7903 

94 
95 
97 

1498.8 

1553.4 
1610.2 

546 
568 
592 

0.57473 
58955 
60484 

1482 
1529 
1579 

7  934 
'      8.081 
'   8.231 

147 
150 
154 

40 


TABLE  II.— Continued. 


V 

Siv) 

Diff. 

A  {v) 

Diff. 

I{v) 

Diff. 

T{v) 

Diff. 

625 

620 

6.5 

8000 
8098 
8198 

98 
100 

lOI 

1669.4 
1731.2 
1795-6 

618 
644 
673 

0.62063 

63696 
65386 

1633 
1690 

1737 

8.885 

8-543 
8.705 

158 
162 
166 

610 
605 
600 

8299 
8402 
8507 

103  1 

105 

107 

1862.9 

I933-I 
2006.4 

702 
733 
765 

0.67123 
68922 
70781 

1799 

1859 
1923 

8.871 
9.041 
9.215 

170 

174 
179 

595 
590 

585 

8614 
8722 
8833 

108 
III 
1 12 

2082.9 
2162.9 
2246.5 

800 
836 
872 

0.72704 
74692 
76747 

1988 

2055 
2126 

9.394 
9.577 
9-765 

183 
188 
192 

580 
575 
570 

8945 
9059 
9175 

114 
116 

118 

2333-7 
2424.8 
2520.2 

911 
954 
998 

0.78873 

81072 

83348 

2199 
2276 
2356 

10.957 
10.154 

10.357 

197 

203 
208 

565 
560 

555 

9293 
9413 
9535 

120 
122 
124 

2620.0 
2724.3 
2833.4 

1043 
1091 
1142 

0.85704 
88144 

90670 

2440 
2526 
2617 

10.565 
10.778 
10.997 

213 
219 

225 

550 
545 
540 

9659 
9785 
9914 

126 
129 
131 

2947.6 
3067.2 
3192.4 

1196 

1252 
1312 

0.93287 

95998 
98808 

2711 
2810 
2913 

11.222 

11-453 
11.690 

231 
237 
243 

535 

-  530 

525 

10045 
10178 

135 
138 

3323-6 
3461.0 
3605.0 

1374 
1440 

1509 

I.OI72I 

1.04740 
1.07873 

3019 
3247 

11-933 
12.183 
12.440 

250 

257 
264 

520 

515 
510 

10451 
10591 
10734 

140 

143 
146 

3755-9 
3914.1 
4080.1 

1582 
1660 
1743 

I.III20 
1. 14486 
I.I7981 

ZZ(^(> 
3495 
3633 

12.704 

12.975 
13-254 

271 

279 

287 

505 
500 

495 

10880 
11028 
11179 

.48 
151 

'53 

4254-4 
4437-3 
4629.3 

1829 
1920 
2017 

I.21614 

1.25393 
T. 29312 

3779 
3919 

4070 

13-541 
13.836 
14.138 

295 
302 
312 

490 

485 
480 

11332 
11488 
1 1648 

156 

160 
162 

4831.0 
5042.8 
5265-4 

2118 
2226 
2340 

1.33382 
I.37614 
1. 42013 

4232 
4399 

4575 

14-450 
14.770 
15.100 

320 
330 
340 

475 
470 

465 

11810 

11975 
12143 

168 
172 

5499-4 

5745-5 
6004.3 

2461 

2588 
2724 

1.46588 
1. 51348 
1. 56301 

4760 
4953 
5157 

15-440 
15-790 
16.150 

350 
360 

370 

41 


TABLE  II. —Continued. 


V 

S(v) 

Diff. 

A{v) 

■ 

Diff. 

I{v) 

Diff. 

T{v) 

Diff. 

460 
455 
450 

12315 
12490 
12668 

175 
178 

6276.7 
6865.5 

2868 
3020 

1. 61458 
1.66826 

!  1. 72419 

1 

5368 
5593 

16.520 
16.902 
17.296 

382 
394 

42 


TABLE  III. 


f) 

(^) 

Diff. 

Tan^ 

Diff. 

e 

(^) 

Diff. 

TanO 

Diff. 

o°  oo' 
o  20 
0  40 

0.00000 
00582 
01164 

i 
582 
582 
582 

0.00000 
00582 
01164 

582 
582 
582 

0  ' 
II  00 

II  20 

II  40 

0.19560 
20176 
20794 

616 
618 
621 

0.19438 

20042 
20648 

604 
606 
608 

I  00 
I  20 
I  40 

0.01746 
02328 
02910 

582 
582 
583 

0.01746 
02328 
02910 

582 
582 
582 

12  00 
12  20 
12  40 

0.21415 
22038 
22663 

623 
625 
627 

0.21256 
21864 

22475 

608 
611 
612 

.2   00 
2   20 
2   40 

0.03493 
04076 
04659 

583 
583 
584 

0.03492 

04075 
04658 

583 
583 
583 

13  00 
13  20 
13  40 

0.23290 
23920 
24553 

630 

633 
636 

0.23087 
23700 
24316 

613 
616 
617 

3  00 
3  20 

3  40 

0.05243 
05827 
06412 

584 
585 
586 

0.05241 
05824 
06408 

583 
584 
585 

14  00 
14  20 
14  40 

0.25189 

25827 
26468 

638 
641 
644 

0.24933 
25552 
26172 

619 
620 

623 

4  00 
4  20 
4  40 

0.06998 

07585 
08172 

587 
587 
58S 

0.06993 
07578 
08163 

585 
585 
586 

15  00 
15  20 
15  40 

0.27112 

27759 
28409 

647 
650 
654 

0.26795 
27419 
28046 

624 

627 
629 

5  00 

5  20 
5  40 

0.08760 
09349 
09939 

589 
590 
591 

0.08749 

09335 
09922 

586 

587 
588 

16  00 
16  20 
16  40 

0.29063 
29720 
30380 

657 
660 
663 

0.28675 

29305 
29938 

630 
633 
635 

6  00 
6  20 
6  40 

0.10530 
II 122 
11715 

592 
593 
594 

0.105  10 
1 1099 
11688 

589 
589 
590 

17  00 
17.  20 
17  40 

0.31043 
31710 
32381 

667 
671 
674 

0.30573 
31210 
31850 

637 

640 
642 

7  00 
7  20 
7  40 

0.12309 
12905 
13502 

596 
597 
598 

0.12278 
12869 
13461 

591 
592 
593 

18  00 
18  20 
18  40 

0.33055 

33733 
34415 

678 
682 
686 

0.32492 
33^36 
33783 

644 
647 
650 

8  00 
8  20 
8  40 

0.14100 
14700 
15301 

600 
601 
603 

0.14054 

14648 

1  15243 

594 
595 
595 

19  00 
19  20 
19  40 

0.35101 

35791 
36486 

690 

695 
699 

0.34433 
35085 
35740 

652 
655 
657 

9  00 
9  20 
9  40 

0.15904 
16509 
17116 

605 
607 
608 

0.15838 
1  16435 

1  17033 

j 

597 
598 
600 

20  00 
20  20 
20  40 

0.37185 
37888 
38596 

703 

708 

713 

0.36397 
37057 
37720 

660 
663 
666 

10  00 
10  20 
10  40 

0.17724 

18334 
18946 

610 
612 
614 

0.17633 
18233 
18835 

600 
602 
603 

21  00 
21  20 
21  40 

0.39309 
40026 
40748 

717 

722 
728 

0.38386 
39055 
39727 

669 
672 
676 

43 


TABLE  III.— Continued. 


.  0 

m 

Diff. 

Tan  6/ 

Diff. 

6 

(^) 

Diff. 

TanB 

Diff. 

22°  Oo' 
22   20 
2  2   40 

0.41476 
42208 
42946 

732 
738 
744 

0.40403 
41081 
41763 

678 
682 
684 

33 
33 

00' 

20 

40 

0.69253 

70245 
71248 

992 

roo3 
1015 

0.64941 

65771 
66608 

830 
837 
843 

23  00 
23   20 
23  40 

0.43690 

44439 
45193 

749 
754 
760 

0.42447 

43136 

43828 

689 
692 
695 

34 
34 
34 

00 
20 
40 

0.72263 
73290 
74330 

1027 
1040 

1052 

0.67451 
68301 
69157 

850 
856 
864 

24  00 
24   20 
24  40 

0-45953 
46719 

47491 

766 

772 
778 

0.44523 
45222 

45924 

699 

702 

707 

35 
35 
35 

00 
20 
40 

0.75382 
76447 

77525 

1065 
1078 
1092 

0.70021 
70891 
71769 

870 
878 
885 

25   00 

25   20 
25   40 

0.48269 
49054 
49845 

785 
791 

798 

0.46631 

47341 

48055 

710 

714 

718 

36 
36 
36 

00 
20 
40 

0.78617 
79723 
80843 

1 106 
1 120 
1 1 34 

0.72654 
73547 
74447 

893 
900 
908 

26  00 
26   20 
26   40 

0.50643 
51448 
52260 

805 
812 
818 

0.48773 

49495 
50222 

722 
727 
731 

37 
37 
37 

00 
20 
40 

0.81977 
83126 
8429 1 

1 149 
1 165 
1T82 

0-75355 
76272 
77196 

917 
924 
933 

27   00 
27   20 
27   40 

0-53078 
53904 
54738 

826 

834 
842 

0.50953 
51688 

52427 

735 
739 
744 

38 
38 
3^ 

00 
20 
40 

0.85473 
86670 
87883 

1197 
1213 
1231 

0.78129 
79070 
80020 

941 
950 
958 

28  00 
28   20 
28  40 

0.55580 
56429 
57286 

849 

857 
865 

0.53171 
53920 

54073 

749 

753 
758 

39 
39 
39 

00 
20 
40 

0.89114 

90363 
91629 

1249 
1266 

1285 

0.80978 
81946 
82923 

968 

977 

^987 

29  00 
29   20 
29  40 

0.58151 
59025 
59907 

874 
882 
892 

0.55431 
56194 
56962 

763 

768 

773 

40 
40 
40 

00 
20 
40 

0.92914 
94217 
95541 

1303 
1324 
1343 

0.83910 
84906 
859'2 

996 
1006 
1017 

30  00 
30   20 
30  40 

0.60799 
61699 
62608 

900 

909 
919 

0.57735 
58513 
59297 

778 

784 
789 

41 
41 
41 

00 
20 
40 

0.96884 
98247 
99632 

•363 
1385 
1407 

0.86929 

87955 
88992 

1026 
1037 
1048 

31   00 
31   20 
31   40 

0.63527 

64455 
65394 

928 
939 
949 

0.60086 
60881 
61681 

795 
800 
806 

42 
42 
42 

00 
20 
40 

1.01039 
02468 
03920 

1429 
1452 
U75 

0.90040 
91099 
92170 

1059 
1071 

T082 

32   00 
32   20 
32   40 

0.66343 
67302 
68272 

959 
970 
981 

0.62487 
63299 
64117 

812 
818 
824' 

43 
43 
43 

00 
20 
40 

1.05395 
06894 
08418 

1499 
1524 

1550 

0.93252 
94345 
95451 

1093 
1 106 
1118 

44 


TABLE  III.— Continued. 


44  oo 

44  20 

44  40 

45  00 
45  20 
45  40 


m 


1.09968 

I-II544 
1.13148 

1. 14779 
1. 16439 
1.18129 


46  00  1. 19849 


46  20 

46  40 

47  00 
47  20 

47  40 

48  00 
48  20 

48  40 

49  00 
49  20 

49  40 

50  00 
50  20 

50  40 

51  00 
51  20 
51  40 


r. 21600 
1.23384 

1. 25201 

1-27053 
1.28940 

1.30863 
1.32823 
1-34823 

1.36863 
1.38944 
1. 41068 

1.43236 

1.45450 
1. 47710 

1. 50019 
1-52379 
I-54791 


Diff. 


1576 
1604 
1631 

1660 
1690 
1720 

1751 
1784 
1817, 

1852 
1887 
1923 

i960 

2000 
2040 

2081 
2124 
2168 


2214 
2260 
2309 

2360 
2412 
2466 


Tan  S 


0.96569 
97700 
98843 

1. 00000 
1.01170 
1-02355 

1-03553 
1.04766 
1.05994 

1.07237 
1.08496 
1.09770 

1.II06 
1. 12369 
1. 13694 

1. 15037 
r. 16398 
1. 17777 

1.19175 
1-20593 
1. 22031 

1.23490 
1.24969 
1. 26471 


Diff. 


1131 
1 143 
1157 

1170 
1185 


1213 
1228 
1243 


1259 
1274 
129 

1308 
1325 
1343 

1361 

T379 
1398 

1418 
1438 
1459 

1479 

1502 
1523 


52  00 

52  20 

52  40 

53  00 
53  20 

53  40 

54  00 
54  20 

54  40 

55  00 
55  20 

55  40 

56  00 
56  20 

56  40 

57  00 
57  20 

57  40 

58  00 
58  20 

58  40 

59  00 
59  20 

59  40 

60  00 


(^) 


Diff. 


1-57257 
1.59779 
1.62357 

1.64995 
1.67696 
1.70460 

1. 73291 
1.76191 
1. 79162 

1.82207 

1-85329 
1.88530 

1.91815 
1.95186 
1.98646 

2.02199 
2.05849 
2.09600 

2.13456 

2.1742 

2.21500 

2.25697 

2.3001 

2.34468 

2-39053 


Tan^ 


Diff. 


2522 

2578 

638 

2701 
2764 
2831 

2900 
2971 

3045 

3122 
3201 
3285 

3371 
3460 

3553 

3650 
3751 
3856 

3965 
4079 
4197 

4321 

4450 
4585 

4726 


1.27994  1547 
1. 29541  1569 
1.31110  1594 

1.32704  1619 

1-34323  1645 
1.35968  1670 


1.37638 
1.39336 
1.41061 


1.42815 
1.44598 
1.46411 

1.48256 

1.50133 
1.52043 

1.53986 
1.55966 
1.57981 

1.60033 
1. 62125 
1.64256 

1.66428 
1.68643 
1. 70901 

1.73205 


1698 
1725 
1754 

1783 
1813 

1845 

1877 
1910 
1943 

1980 
2015 
2052 

2092 
2131 
2172 

2215 
2258 
2304 

2351 


45 


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S"^'    '.1 


V2^ 


^2M- 


i\oM    ST  184S 


<'"?vB5lB 


Apr'59BBl 


V-V 


MAR  26  1989 


OCT    3.)    1943 


FEB    1  1944 


S^ 


^ea 


JUL    16  li-40 


Ak^m^-^ 


^■Ttj^ftl 


^^ 


2e3w.»'^  * 


"itV 


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SENT  ON  It  I 


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